San Fernando Valley State College
A COMPARISON OF SEVERAL
ll
MEASURES OF INVARIANCE
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Arts in
PsyQhology
by
Anson Justin Levine
July, 1967
The thesis of Anson Justin Levine is approved:
San Fernando Valley State College
July, 1967
ii
Acknowledgments
I should like to acknowledge the efforts of Bryce C.
Schurr who, despite many other burdens, generously gave his
own time to help me expedite the data processing.
Processing of the data could .not have been
accomplished without the invaluable assistance and
facilities provided by Western Data Processing Center,
University of California, Los Angeles.
Finally; I wisp to express my appreciation and grati_tude to my wife who worked very hard to make this paper
possible.
iii
Table of Contents
Page
Acknowledgments
iii
Abstract
Introduction
viii
..
1
Method
5
Results
22
Discussion
73
Conclusion
85
References
88
Appendix
89
v
List of Tables
iv
LIST OF TABLES
Page.
2.
3.
4.
Identical Scores and Procedures
Modified Coeffictent of Invariance
25
Identical Scores and Rotation
Coefficient of Congruence
Coefficient of Invariance
27
Identipal Scores and Rotation
Modified Coefficient of Invariance .
28
Original Scores and Permuted Scores
30% Rows
Coefficient of Congruence
30
. . . . . .. . .
5.
6.
Original Scores and Permuted Scores
30% Rows
Modified Coefficient of Invariance
.
Original Scores and Permuted Scores
60% Rows
Coefficient of Congruence
. .
...
32
. .
33
. . . ... .
35
. . .
'7
0
8.
0
Original Scores and Permuted Scores
90% Rows
Coefficient of Congruence
Original Scores and Permuted Scores
30% Columns
Coefficient of Invariance ( z)
0
9.
0
0
. . .
Original Scores and Permuted Scores
30% Columns
Coefficient of In variance (Z*)
. .
10. Original Scores and Permuted Scores
30% Columns
Modified Coefficient of In variance
0
39
. . .
40
.
41
. .
43
. . . .. . .
44
11. Original Scores and Permuted Scores
60% Columns
Coefficient of Invariance ( z)
. .
12. Original Scores and Permuted Scores
60% Columns
Coefficient of In variance ( z *)
v
0
...
. . .
0
0
0
0
Page:
13.
Original Scores and Permuted Scores
90% Columns
Coefficient of Invariance (Z)
. . . . . ..
14.
15.
Original Scores and Permuted Scores
90% Columns
Coefficient of In variance (z*)
....
47
Original ·scores and Permuted Scores
30% Rows and Columns
Coefficient of Congruence
....
51
Original Scores and Permuted Scores
30% Rows and Columns
Coefficient of In variance {Z)
. . .. ...
53
Original Scores and Permuted Scores
30% Rows and Columns
Coefficient of Invariance (Z*)
54
Original Scores and Permuted Scores
30% Rows. and Columns
Modified Coefficient of Invariance
55
Original Scores and Permuted Scores
60% Rows and Columns
Coefficient of Congruence . . . • .
56
Original Scores and Permuted Scores
60% Rows and Columns
Coefficient-of Invariance (Z) . . .
58
Original Scores and Permuted Scores
60% Rows and Columns
Coefficient of Invariance (Z*)
59
Original Scores and Permuted Scores
100% Rows and Columns
Coefficient of Congruence . . . . •
61
Original Scores and Permuted Scores
100% Rows and Columns
Coefficient of Invariance (Z) • • • • • • •
62
Original Scores and Permuted Scores
100% Rows and Columns
Coefficient of Invariance (Z*)
63
. . .
16.
17.
19.
i20.
121.
22.
'23.
24.
46
.
vi
Page
25.
26.
Original Scores and Permuted Scores
100% Rows and Columns
Modified Coefficient of Invariance
64
Original (Pre) Scores and Post Scores
Typical Study
Coefficient of Congruence
Based on Principal Components
66
Original (Pre) Score.s and Post Scores
Typical Study
Coefficient of In variance ( z)
Based on Principal Components
68
Original (Pre) Scores and Post Scores
Typical Study
Coefficient of IJ1variance (Z *)
Based on Princ±pal Components
69
Original (Pre) Scores and Post Scores
Typical Study
Modified Coefficient of Invariance
Based on Varimax Loadings
70
.......
27.
. . . . . . .
28.
.......
29.
. . . . . . ...
30.
. :31.
32.
Original (Pre) Scores and Post Scores
Typical Study
Coefficient of Congruence
Based on Varimax Loadings . . . . . . • .
.
72
A Comparison of Several Measures
of Invariance
Summary of Analyses •
87
"Varimax Solution for Eight~Physical
Variables"
• • .
.. . . . . • . .
90
vii
ABSTRACT
A COMPARISON OF ·SEVERAL MEASURES
OF INVARIANCE
by
Anson Justin Levine
Master of Arts in Psychology
July, 1967
Criticisms have
b~en
raised against factor analysis
1
for lack of objective and reliable methods for comparing
•
'sets of factor loadi~gs.
In an attempt to meet such
criticisms, three measures of invariance or similarity of
:factors have been offered:
l
the coefficient of congruence,
I
[the coefficient of invariance and the modified coefficient
I
i
jof in variance ..
I
The present study develops criteria for an adequate
measure of invariance and compares each of the measures
1
!with these criteria.
i
The five criteria presented are
:based on considerations of what value a good measure of
'invariance should yield when applied to five transformations
of an initial data matrix:
procedures,
permutation,
(2)
(4)
(1)
Identical scores and
Identical scores and rotation,
Column permutation, and (5)
viii
(3)
Row
Row and
column permutation.
The results showed that the coefficient of congruence
did not meet the specified criteria for an adequate measure
of invariance for any of the cases involving permuted data
matrices.
The coeff~cient of invariance met these
criteria; however, it did not produce a unique solution.
The modified coefficient of invariance did not meet the
.
originally·specified criteria in those cases involving
'
identical scores and varimax rotation, or row permutation.
When it was discovered that the varimax factors were not
orthogonal, alternative criteria for an adequate measure of
invariance were developed.
These new criteria reveal that
the modified coefficient of invariance is a more adequate
measure of the similarity of sets of factors than either' of
the other measures.
Further results are presented for the
modified coefficient and the other measures, for a typical
;(pre-post) application of measuring the invariance of
factors.
'
ix
Introduction
i
I
II
One of the most .important· requirements
of the
.
jscientific method is that res.earch findings can be
/replicated.
That is, results should be achieved in such a
!
'
'way
that another investigator repeating.the exact
procedures followed in the original investigation can
arrive at comparable· results on another occasion.
The replicability require.ment has been exceedingly,
difficult to meet in studies employing factor analytic
techniques because objective measures ·of "invariance" have
·not been available; i.e., measures of the extent to which
one has obtained comparable findings,
Henrysson, 1957;
Thu~stone,
1947).
(Harman, 1960;
Thus, although the
same procedures could be followed in two studies, it was
not possible to objectively compare the results.
For example, one might be interested in comparing
results obtained at several different colleges at which
the same standard entrance examinations (e.g., SAT, SCAT,
ACT) are administered.
Suppose that each college,
employing the same factor analytic procedures,
independently analyzed its data into ten "composite tests"
or factors.
The problem of invariance is how similar are
the factors or clusters obtained at the ditferent colleges.
Thomson (1951) examines the reasons for employing
1
2
factor analysis and indicates that perhaps the most
fundamental advantage of the procedure is the independent
parsimonious factor structure it provides to interpret or
describe data.
To illustrate his point Thomson provides an
I
!example of a man who .can exchange five cows for so many
i
.
:sheep, so much cloth, a plow, etc. i however, the man soon
biscovers that such exchanges become much simpler if the
~ransactio~s are conducted for doll~rs and cents instead of
merchandisee
Thus money can be considered a "factor" in
this example.
Clearly, it would not be.possible to make
·use of the "money factor 11 if money did not have a
'consistent value from one transaction to the next.
With
'
respect to psychological investigation, one might infer
l
'that those factors with consistent, reliable, invariant
!
!interpretations are likely to be the most useful in
I
and describing
behavior.
Ihnderstanding
.
.
I
i
I
Until recently, objective methods for comparing
~actors have not ~een available.
!
In the absence of an
iobj~ctive measure, investigators have had to rely on
:"inspection" to compare factors.
As a consequence, factor
I
'
;analysis
has been repeatedly criticized for lacking
i
-~onsistency
and objectivity.
Certainly, results achieved
!
~y_the method of "inspection" are difficult, if not
i
I
!impossible, to
repe~t.
Thurstone (1947) states,
".~.
if
l
I·any part1.cu
.
1 ar method of factoring is specified in
sufficient detail, then two authors would generally get
the same numerical values in the factor matrix, but
tha~
3
sort of computing agreement is not of theoretical or
scientific stgnificance (p. 363)."
Recently, several
:methods have been suggested which appear to provide a
i
measure of similarity of factors which is of theoretical
iiand
I
I
scientific signif·icance.
These methods have been presented in a theoretical or
, istrictly methodological framework without the necessary
:empirical results of their application to actual data,
(Wrigley and Neuhaus, 1955; Pinneau and Newhouse, 1964; and
Pinneau, Schurr, and Levine, 1966).
In some cases authors
have presented new methods of measuring invariance and
:~ccompanied these presentations only with a practical
Fxample of their method, while others chose to present
!
pypothetical examples demonstrating the deficiencies of
I
.
~ther
measures
in comparison to their own..
i
. .
In any case,
the investigator who is faced with the problem of measuring
I
I
the degree of relati9nship between two sets of factors has
i
I
ho reference which contrasts the several methods of
1
.
~easuring invariance with actual criteria of what an
II
~dequate
I
measure should yield.
The intention of the present stUdy is to develop
criteria for an adequate measure of invariance and to
.determine the extent to which a number of different methods
of measuring invarianc'e meet these criteria.
I
1
The methods which wiil be compared consist of the
coefficient of congruence, Wrigley and Neuhaus (1955),
coefficient of invariance, Pinneau and Newhouse (1964), and
4
the modified coefficient of invariance, Pinneau, Schurr,
and Levine (1966).
There are a number of designs to which a measure of
invariance may be applied.
Under one design the subjects
'
of the two studies ar~ fixed but the variables are
i
'
different.
A second design
!
put fixed variables.
'
~ay
·involve different subjects
Still another design may involve
!i
'fixed subj~cts and fixed variables on two occasions.
The
term "fixed" is used, as defined by Pinneau, Schurr, and
Levine (1966), " ••• to imply constant subjects or
variables, listed in the same order for both sets of data
· (p. 2) • "
Generally, in any ··of the above designs, the term
:n invariance" refers to the relationship between the
i'
·!factors of the two studies. · However, Pinneau and Newhouse
J
(1964), show that the conditions which are necessary to
!fulfil-l the requirements of the coefficient of congruence
are different from those required for the coefficient of
invariance in all but the fixed-variable, fixed-subject
1
design.
Consequently, the meas'+res can be compared only
!for this case.
Method
i
In order to adequately compare the methods of
I
!measuring invariance for each of the conditions, eleven
. !separate correlation matrices were factor analyzed
\
'!
accorditig to the principal-components solution developed in,
Harman (1960).
Each principal components solution was
rotated according to the varimax criteria, Kaiser (1958).
'In these computations the methods remained identical for
I
\each analysis.
'
·The diagonal elements of each correlation
_:matrix were set equal to 1.0 and the criterion employed to
i
!
· Jstop extracting additional factors was that of lambda (I.)
r
Iless than
!
1. 0.
Before the three measrires of invariance are defined,
the basic equations of principal components analysis are
,presented.
In these equations 'the initial correlation
I
!matrix R_represents the degree of relationship between the
I
•
-\original scores which have been standardized (Z), i.e., the
i
!mean and standard deviation of each column of the matrix of
-\original scores is made zero and one, respectively.
i
.
The
lequatioris w~ich follow repiesent two sets of data factor
!analyzed us1ng the procedure of principal components; in
I
.
lboth instances all of the factors necessary to reproduce
j
1
the correl~tion matrix are ~etained.
Where L and L* are matrices of factor loadings and
5
6
.s and S* are matrices of factor scores,
z
1
N
I
z =
R
=
z =
s
Ll
s
= z
L
L
1
Ll
Z*~
N
-11
Z*
=
R*
=
Z*
=
·s*
L*
S*
=
Z*
.!Figure 1 presents the above relationships.
\.
t
L*
L*l
I
L *-11 •
The measures of
.
I
'invariance. may now be ·defined for the case of fixed
variables and fixed subjects.
As noted earlier, the method of inspection involves a
!visual comparison of the
loa~i~g
of each variable on each
'"composite variable" or factor in one study with the
i
i
·'!study.
Since this procedure is highly contaminated by the
lbiases and expectations of the investigator, it cannot be
!
/regarded as a scientific method. · Hence, the present paper
I .
!will only deal with the three objective measures of
I
Jinvariance.
I . Three Objective Methods of Measuring' Inva:ria:nce
jcoefficient of ·con"grue·nce ·
I·
The coefficient of congruence may be considered as a
I
!matrix 6£ correlations, R , between two sets of factor
c
scores, s and S*:
[1]
Since s
= z
R
c
. -1 I
L
and S*
=
1
s~
N
=
Z*
S*.
L*-1'
by substitution
'
~
7
1. •. .
1
N
,.
1.. . M
N
~~
k~
tt:l
M~
1
z
=
N....._._ ___,
1
.
:
M
1.
tJ
FIG. 1.
·,·}
• 1
M
'
1..• Q
:~1
Q.
-"
1. •• Q
:le • • M
1
s
N
1[J·•. 1·~
1... M
'
z
=
l\L- I
N
Equations for Factor·
and Factor Scores
Loadin~s
11
8
cohgru~nce
the coefficient of
=
1
becomes
z'
N
Z*
The matrix of intercorrelations between'the different
i
I
!z matrices is given by· t:.he equation
I
I
I
1
,I
N
z'
=
Z*
'·' i
hence, by $Ubstitution,
=
[ 2]
·coefficient of Invariance
The coefficient of .invariance, Rrv
I'
z
is the matrix of
icorrelations between two sets of 'factor scores, S and S*,
· Z
'[where
I
I
iI
·s
z
=
L-1' and S*
z
'
I
1
=
Rrvz
=
S'
N
1
=
S*
z
L-1
N
I
L*-1'; thus
z
z'
z
I
iThe intracorrelation matrix for the first set of variables,
I
IR11 =
~
Z'
Z, is a factor in this equation; hence, by
I
j'
I
I
l subst~ tut~on
I
!
I
Rrvz
=
L-1
Rll
=
L-1
L
=
I
L'
L*-1''
L'
L*-1'
L*-1'
9
-· L 1
[3]
1
L*-1 •
Alternatively, the coefficient of invariance, RIVz*' may
.be computed using factor scores based upon z·*.
!
I
I
jif Sz*
=
Z*
L-1~
and S*
I
I
R
IVz*
I
,I
L*-1 I ,
Z*
-1
=
=
s~
S*
N
Z*
1
L-1
N
Z*l
Z*
L*-1 I .
The intracorrelation matrix among the second set of
·:variables, R2 2
!
1
= N
Z*l
:RIVz*
-
t""l
R22
=
L-1
L*
L*l
=
L-1
L*
I
=
L-1
L*
z~,
appear~
as a factor in this
i
~quation;
hence,
I
I
i
·!I
I
I
I
I
II
II
I
II
I
I
[4]
L*""l'
L*-11
I
I
i
:Modified Coefficient of Invariance
The modified coefficient of invariance,
R~V'
/the correlation between two sets of factor scores,
is also
so
and
./s*o:
I
'
=
1
N
so•
S*o,
i
'In this case the matrix of factor scores is computed using
'a matrix of orthogonal standard scores (Z 0 ) , i.e., the
10
subjects' scores on one variable are independent of their
scores on each of
th~
other variables.
factor scores, S 0 and S* 0
The two sets of
are developed using the same
,
zo
I
'matrix and two different sets of factor loadings:
= zo
Iland
zo
=
1
L-1*':
i
)By substitution_, the equation for the modified coefficient
\._· l
.of invariance becomes:
R~v
=
1
zo'
N
zo
1 zo• zo = RI = I, since the
N
•
perfectly cortelated with themselves and
!In this equation,
I
I
~ariables
ar~
l
i
Uneerrela.ted with each ether J hemee,
I
1
[5]
I
[6]
R~v
=
I
=
The three methods that have been proposed to measure
.,iinvariance all center around the issue of how similar the
Meights
or contri.butions of the variables to each factor
.
I
iof one study are to the loadings or weights of the same
variables on each factor of another study.
Ideally, each
lof these methods of measuring invariance represents an
!attempt to quantify the degree of relationship between the
I
!two
sets of factor. loadings.
I
.
In each instance the
!
~procedure may be viewed as the correlation between factor
lscore~ based on th~ two sets of loadings, one from each'
11
study; hence, the letter !'R" is used in the designation of
all three measures.
Conceived in this manner, the differ-
'ences between· the measures involve the method used to
I
.
I
.
!compute the factor
I
sco~es.
A letter subscrl·pt ·is used to differentiate the three
~easures
and.to indicate that the relationships are
\ factually correlations between two sets of factor scores
i
rather than two sets of standard scores.
However, the
three measures of invariance yield different results
under certain conditions because the factor scores they
employ are based on different Z matrices.
I
Deciding which
;
:z
I
matrices are to be used becomes a dec.is:i.o:n as to which of
I:the . thret\l m@a!llure§ mmrt E~ati§faetorily
.
. m@et eri teria of an
'!1adequate measure of invariance.
I
Ii
I
Five Conditions for Measuring Invariance
The following conditions have been developed in order
J
:to establish criteria for an adequate measure of invarilance.
I
.
These conditions actually represent data matrices
~hich are used to compute the appropriate factor scores for
ieach of the three objective measures of invariance.
I
These
I
!five sets of data represent purely artificial circumstances
'i
Jin which the nature of the relationships, can b~ predicted
~~
priori.
These data are important because they represent
I
:a condition where criteria for a good measure of invar'iance may be specified before the invariance of the two
sets of factors are computed.
Thus, it will be possible
12
~
to state the desired result with each case and to see
which measure most satisfactorily meets the specified
criteria •
''
!
. Rather than focus on mathematical_ relationships
I
!between the various ~easures, the present research tests
i
!the measures using actual data.
!
The advantage of employ-
I
. jing actual data rests upon the expectation that the results
' ·- . II
!may reveal additional relationships which lead to increased
..
comprehension of the advantag~s and disadvantages inherent
;
;in employing one or another of the measures of invariance.
;
i
i
I
;criterion 1:
iI
Identical Scores and Procedures
Two sets of identical scores yield the same factors;
ithus, an adequate measure of invariance should show that
I
'1
I
.
!matching factors of the two sets are perfectly related •.
I
In this. case of identical scores and procedures, each
jscore of each individual on the variables in the pre test
lare identical
wi~h
the scores of each individual on the
,same set of variables on the post test.
The factor
ana+ytic procedures used are identical for both studies.
'
.
!In order to accomplish this comparison, instead of using
I[the
actual post test scores, a duplicate set of pre scores
I
!are substituted for the post scores so that the comparison
I
!
lis between identical. f·actor scores ·based on identical sets
I
When z = Z*, L should equal L*.
iof data.
I
I
The value of this example rests on the fact that two
sets of identical
scor~s
should yield identical sets of
factor loadings if the methods of determining the factors
13
~
are the same for each set.
Thus, an appropriate i:hvariance
measure should show a perfect matching.
Since the measures
being considered are correlation coefficients, each should
I
i
~ield
a value of 1.00 for matching factors.
II
Criterion 2:
I
Identical Scores and Rotation
One may obtain a second set of factor loadings L*
i
i
'
''by post multiplying the principal compon~nts loading
matrix by an orthonormal transformation matrix T; i.e.,
L*
L
=
T.
Therefore, an adequate measure of invariance
should indicate that L* is just an orthonormal transforma'
:tion
of L.
I
!
1
In the present study the three measures of invariance
~tilize
the principal <;:omponents factors and the same
·
>!
!factors rotated by the varimax method developed by Kaiser
!
j(l958)
·~
The varimax method utilizes an iterative procedure
Ito obtain the orthonormal transformation of the row normallized loadings.
T.he scores on the variables are not
!
modified in any way; i.e., the standard scorematrices in
I
both pre and post (rotation) test are
sti~l
equal.
i
I
:criterion 3:
i
Permuted Rows
An adequate measure of invariance should show that
I
I
1two sets of· factor loadings are perfectly related when the
!factor loadings are identical.
Such conditions are offered
i
~Y Pinneau and Newhouse .(1960)
as evidence of the superi-
!
'ority of their measures over those presently available.
The following development demonstrates that permuting rows
14
does not modify the correlation matrix, and hence does not
modify the .factor loadings.
If p is the permutation
~
matrix, Z*
I
p
=
I
I
I
R*
!
I
I;and,
z
1
=
N
=
N
=
N.
=
N
=
N
Z*'
Z*
by substitution,.
I
I
R*
,I
.. I
I
[P
z] I
1
zl
p1
1
zI
I
1
1
z·
p
p
z
z
zI z
-,
= R.
I£ th® sam® proe®dlif'fl!':J ar® ®mploy®rl to £aetor the
·!correlation matrices produced by
lcorrel~tion
z
and
z*,
and these
matrices are identical, then the two sets of
I
[loadings should also be. identical.
i
I
Three cases .of row.p~rmut!3.tion are presented; for
ieach of these cases a d1fferent number of rows are inter!
!changed. This procedure is followed in order to systema1
!tically increase the amount of change in the original
I
[matrix of standard scores.
Thus, ·it is possible to
i
!ascertain the extent to which these modifications-effect
i
I
ithe different measures of invariance.
!
Ideally, the
thr~e
jmeasures should continue to yield the same coefficients
lif the sets 6f factor loadings remain identical.
!
.
In the first case 30% of the 274 rows of the original
pre test matrix of standard scores (Z)
~re
interchanged
15
randomly.
The resulting matrix Z* is a row permutation of
the original set of standard
scores~
In the second case
:60% of the rows of Z are randomly interchanged.
i
I
190% of the rows of
i
z
are permuted.
Finally,
These three cases
lshould indicate what ·happens to the measures of invariance
!when the relationships between the rows of the permuted·
I
.
, · 1matrix become more and more disparate.
:
Since'the relationship between the variables is not
disturbed by permuting the rows of the data matrix, the
intracorrelation matrix for each of the three permuted
matrices should be the same as that obtained for
z.
As
stated by Pinneau, Schurr and Levine (1966), an appropriate measure of invariance should reflect the degree of
.i
borrespondence between the two sets of factor loadings.
!
fhus t~e criteria for an adequate measure of invariance
in this case ·is an identity matrix.
1
I
I
:criterion 4 •
Permuted Columns
Disturbing the order of the scores on the variables,
i.e:, the columns of
I
z,
results in two sets of factor
I
iloadings which· are not identical.
~easure
Therefore an adequate
of invariance should show that changing the order
l
bf the columns modifies the relationships between the two
I
'sets of factors.
Instead of interchanging a row of Z to form Z*, in
I
this case all of the scores for one variable or column are
randomly reassigned.to new column locations in Z*.
The
16
relationship between the columns of the original matrix and
the permuted column matrices are gradually reduced by
randomly 'interchanging 30% of the original columns (14
:columns), 60% of the original columns (28 columns), and
!90% of the original columns (41 columns).
I
I
I
Z*
=
z
p
R*
=
1
N
Z*l
R*
=
1
N
[Z
P]
pi
z
.I
, I
'
i
For this case,
Z*;
by substitution
=
=
!i
i
I
·[
1
I.
N·
p1
z
I
z
p
p
p.
R
Since permuting tl).e rows and/or columns of a matrix
'
~oes not effect the value of the numerical entries, only
I
/their location; the preceding equation shows that R and R*
:contain the same numerical entries.
The values appearing
!in a given row and column are the same for only those
leases in which unity appears in the principal diagonal 6£
1P.
·Thus, the intracorrelation matrix of Z is not equal to
lhat of Z*.
I
isubsti tution,
!
I
If R*
=
R*
R
P1
=
p
=
L*
P and P
L
Ll
L
=
L* then, by
p1
)
i
L*'·
I
!Therefore, it may be seen that although the matrices of
1
loadings contain the same entries, the loading values are
attached to different variables; hence, the factors are
..
17
no longer the same.
If the loadings for the variables have
changed, then the measures of invariance of the two sets
of loadings should indicate that each factor has changed
;even if the two sets of factor scores are perfectly correi
I
ilated.
I
Another approach, Pinneau and Newhouse (1964), to the
, !problem of factor similarity stresses invariance of factor
'scores rather than in variance of loadings.
According to
this view, the results of a desirable measure of invariance
should yield an identity matrix between the two sets of
;factors (original scores versus column permuted scores).
;
!
)For example, if the variables of the two studies have
i
lcri~f@rcult netl'ftfitllij, . and §cor®!§ en (m.® of the faeter§
'i
i
W{i}f'@
.
ihighly related to scores on a factor of the other study, an
i
!investigator might not wish to dismiss this relationship
i
.
.
merely because the loadings on the factor of the second
i
!study are attached to variables with different names.
I
I
I
'Criterion 5:
!,
Rows and Columns Permuted
. The important feature·of the example created by permu-;
I
'
iting both rows and columns simultaneously is that as more
I
.
i
!and
more rows and columns are interchanged,
this case very
!
.
.
i
iclosely.approximates the circumstance in which there are
I
Ionly
i
/
chance relationships between Z and Z *.
In a typical study within the fixed-subject, fixed-
!
variable design, scores are collected for the same subjects.
on the same variables on two different occasions.
In the
. 18
case of permuting the rows.and columns of one
'form
z
~o
matrix
Z*, again there is different data for each matrix of
.\"
.standard scores •.
I!100%
.I
In the row and column case, 30%, 60%, and
random permutations of both rows and columns should
.
.
!indicate the effect o:f ·the· gradual: distortion introduced
.~etween
I
.
.. ·
the original
Z:matrix and the row and column per·.
.
\_'·~uted ·matrices
'·
. (Z*). : In the 100% case, when all rows and
- i
coluinns have been randomly permuted, only by chance does
.•
z ..
=
i
If these matrices are factor analyzed, then
i l.J
z~
l.J
:
I
and Z*
=
S*
z
=
s L
L*. where the :relationship between L and L*
I
"1,
•
•
and between S and S* is not readJ.ly evJ.dent. Indeed, as in
I
.. I
-'the other cases, an adequate measure of invariance should
. '· I
,.
...
. ',:'
~
'I ·. ·
· directly reflect the comparability of L and L*.
J
Typical Study
The typical study represents the usual case in which
pre and post data is available on the same subjects for the
kame
variables.· In·such i11stances the numerical entries in
I
I.
~oth th~
I
~atrix
post Z matrix and the
resulti~g
factor loading
are different from those computed from the ·pre test
i
pat a.
II.
I
The·. artificial rearrangement of scores on subjects, on
.
variables, or on subjects and variables presented in the
!I
.·.
preceding sections has . a direct bearing. upon the appropriate.measure of invariance for the typical study.
Measures of invariance which·do not meet the specified
19
criteria for the other cases will not satisfactorily
measure the relationships between the factors in the
·•
;typical study.. Hence, .an adequate measure of invariance
I
!should be the measure which most adequately meets the
\criteria
established.f~r
the previous cases.
Thus, the
!typical study is used only to illustrate an application of
·
·
. !leach o f th e measures o f 1nvar1ance.
'
t
I
l
!
Data Description
The data for the analysis consists of two sets of
·:responses (pre and post) of 274 subjects to a 46-item
:attitude survey.
1
The subjects were trainees who partici-
;pated in a two-week Head Start training program designed
i
'ito prepare them to become assistant teachers or teacher
I
I
!aides.
I
I
-
The attitude survey consists
of 46 statements with
.
ifour possible answers ("strongly agree," "agree," "disagree
i
land "strongly disagree") for each statement or item.
~articular
set
i
oi
data was selected because of several
!important considerations:
1 .
This
the size of the sample of sub-
.
!jects and items, the fact that such data are typically
I
!analyzed by factor analytic techniques (in fact, results
I
.
!of
factor analyses of other adaptations of this attitude
!
I
!survey are available), and most important, these data
I
~---------
!1
These data were collected by the Office of Economic
iopportunity Training and Development Center, Evaluations
Section, and have been made available to the writer by the
Director, Dr. Donald R. Thomas~
1
20
fulfill the test-retest condition necessary to compare two
sets of factors extracted from data collected on the same
subjects and the same variables on different occasions.
The procedure which is followed to rearrange or
!permute the scores of. the original Z matrix involves
i
!randomly interchanging entire rows (or columns) of
z
with-
., ·lout changing the original row or column headings. This is
1
achieved by pre (or post) multiplying the original data
matrix by a random permutation matrix.
Summary of Criteria
···.
i
;
The following criteria have been developed for an
i
jadequate measure of invariance:
'i!
1.
In the case of identical scores and procedures the
measure should yield an identity matrix.
2.
For identical scores and rotation the result
should be a transformation matrix.
3.
The criterion for an adequate measure of
invar~
iance when rows are permuted is, as in the first
case, an identity matrix.
'
4.
When columns are permuted, an adequate measure
should not yield an identity matrix; rather, it
should reflect tDe inpreasing disparity·between
the two sets of factor loadings as more and more
columns are rearranged.
5.
Row and column permutation should indicate that
the two sets of factors are not identically
21
related and that the relationship between the
load~
ings decreases as the amount of permutation
increases.·
!
!
.I
The measure which satisfactorily meets the criteria of
I
ian adequate measure of invariance in the artificial cases
!should most directly reflect the relationship between the
I
, itwo sets of factor loadings in the typical study.
I
..
Results
i
The following section presents results achieved for
I
lthe three measures of invariance under each of the condi-
1
' ~ions described earlier.
These five conditions represent
;artificial situations for which criteria for an adequate
measure of invariance were developed prior to computing
the results for each measure.
'
I
•
•
In addition, for several
'
icases results are presented which exceed the scope of the
-•.
;original criteria; these additignal resYlts are presented.
:because they merit consideration in selecting an appropri-
,I
late measure of the similarity or invariance between two
!sets
of factors.
I
Results are also presented for a typical
!application of the measures of invariance.
In order to .distinguish the two sets of factors which
iare being compared, a Roman numeral without a prime will
1represent a
facto~
from the first set of factors, e.g.,
I
:Factor I of the pre test scores.
I
A factor from the second
lset o~ factors is denoted by a Roman numeral and a prime,
·r·g., Factor III' of the post test scores.
I
i
I
Identical Scores and Procedures
The criterion for an adequate measure of invariance
Jfor the case of identical
z
scores, correlation matrices,
factor loadings, and factor scores was
22
~n
identity matrix.
23 ..
Although differential criteria were not established with·
'·,,
·'
'respect to unrotated versus rotated.factor solutions, the
,results
presented below indicate that additional consider-
I
I
· 'ations arise when two identical varimax ·solutions are
I
..
· Jcompared.
I
I
Unrotated Factor Loadings
. Comparing factor scores computed from two identical
•'
.
principal-components factor loadings show that the coeffi.
co~grue~ce,
cient of
.
the coefficient of invariance, and
the modified coefficient of inva·rianc~ result in an identitY,
matrixG
Hence, for principal component factor scores, all
'
these measures meet the criterion of an adequate measure of
.l.nvariance.
:
1-'
i
Rotated Factor Loadings
I
I
Rotating the two identical principal components
i
l~ad-
I
!
ings according to the varimax criteria produced identical
I.
..
~arimax factor loadin_ gs which were used to compute the
I .
.
appropriate sets of factor scores required by each of three
I
.
,
measures of invariance.
I
I
I
Only two of the three measures, the coefficient of
I
.
I
.
.
.
congruence and the coefficient of invariance, met the
.
·.
.
priteria'established for an adequate measure of invariance;
I
!
i.e., both yielded an identity matrix.
'
...
I
Since :the modified coefficient of invariance did not
I
result ·in an· identity .matrix, the results for this meas·ure
require more detailed presentation •...· Table 1 presents the
•
24
obtained matrix of coefficients.
These results show unity
in the principal diagonal indicating that each factor in
one set is perfectly related to itself in the second set,
:i.e., Factor I
i
!etc.
=
Factor I'; Factor II
=
Factor II',
Table I also indicated that Factor I is related to
i
[Factor II', III', IV', etc.
I
Similar relationships are
I
\ , !also found for all of the other factors.
Thus, the modi-
. i
'fied coefficient of invariance indicates that the same
two factors are perfectly related; and that these factors
:are also interrelated with the other factors.
For example,
lthe correlation between scores on Factor I and those on
I
~actor II' is ~.290~.
I'
lity of the varimax factor loadings.
If varimax factor
loadings are orthogonal, then the sum of the crossproducts
1
I
!of factor loadings for any two factors should equal zero.
!Similarly, in the case of principal components the sum of
the
crossproduct~
also·be zero.
of the loadings on any two factors should
A comparison of Factors I and II for the
I
!varimax solution shows the sum. of the crossproducts to be
I
\1.3159; the sum of the crossproducts of the same factors in
I
lthe principal components solution is . 00005.
Thus, the
I
!varimax solution did not result in orthogonal factors.
!
~Since this result has implications for other varimax
Jsolutions, the sum o.f crossproducts for the eight physical
I
I
!variable
problem presented·in
Harman (1960, p. 304-305)
I
.
.
i
was computed.
Table 32 in the appendix shows that the
25
r--·
····-----------·-----1
TABLE l. ····-·---------- -----------------·
IDENTICAL SCORES AND PROCEDURE:S. _
MODIFIED COEFFICIENT OF INVARIANCE
'
'·-' IL---~-------------t
•.'l~_a'l.•.:-.0n...29.'19
... .::o.~r>.Tf!.'l. __.,n~!l.!.a_,. _____Q.JH~~------ _____-:o_. ~.21l! ___-o.csb_l! ___ -.<> !_1_326 ____ -_(!.all co
o .026~
·
-I" .I 0 0,
0;>4S - n. CB91l -11.0450 ·
. ------------------------
.
.
·- --,---------------------------------------------------------------------------- ·--- ---· . -..
--------~--=-~"~·LZ~9~0~9~~J.nnno
-n.JJ1A
o.nn4B
-0.15~5
-o.oqo;g
-n.0994
-0.1265
-.Q.1H!l~ __::o_.~7_7b
-0.1010
O.Oto2l
-o.ouo
-0.153"
-0.1'?_71l_____<!~.l1"_5~-- .
___________.,J1.'!7 !t'i ___ ,_Q.~5J>'i _____ l,_O.!'!l!l. __ _--:o_._(U 1.8 ___ Q. 0.88 5
n.nlto9
o.t~7?
o.0192 -0.0556
-o. 0169
-o .1099
__ 9_.1683_
f·------- . _____ ._,._0~011!9. .. "'~~0.9.5.9... ..-::n..033.6 .... 1. 0000 ... :-0.2004
.0.0602
-0.08lt2
-0.0613
1.0'l00. -0.1030
-0.0431o
-0.0576_ .. :-0.0517_.
-0.0093
-().0528-- :-0._1263.
4
n.'l7:>'1
-~-o
-0.1651
-0.07to6
.tz.u __ ,-_o~1Aa.r. ___.,.o. 016 "---- o.0602.
-n.n1zo;
7
-n.r."~29
-o.n9J7
o.o~oao;
-o.o753
· -o.o"77
.
.-:0.0862... o-!l~lJ.16. .. ~0~1 C9'L..c:O .• OB42 .. -0.1216
o.n116
o.ot45
o.nzoz
o.ooo9
- -- --- ------------------------------------------------------ -------- --- -----.
-0.10 30
1.0000
-0.0431o
-0.0093
A
------~-~~1326
-o.~Jtry
".1b~0 •. 0613 ___.,_0 •.0.035
, -o.n1A4
:-0.1715
O.Ol76
-~n.n9~o ... ,-.D.l5.34.
-n.n777
·---I 'I
n.o111
--
-0.0058
,.,0.1979 ... C!.09a5
n.tC46 -0.0482
-0.0576
-0.0173
-0.0511
-0.121>3
0.0.31)2
-0.0497
-0.0325
0.0336
-0.0381o
o.Oio23
0.0455
n.oo69 ·-o.n99B
-0.1Q~t,
-n.1on6
·t.nnnn
-1".1118
-o.n,19
0.0149
o.OC57
o.072C
-0.0"17
-:o. Ot,5 7
.. ~~~-~~~~~-~--1-~c~-~----ij.
-0.0777
0.0752
i
·--·---·- ----------------·----i
1"7 --------------------------------------------· ------------ ---------- "·- ------·-· . ________ (l,_O.~_'t'i_____ ft.':'_Oie_B ____ !'.,U:U___ ::-_Q ,_')!1_29__
-n.n,19
t.no"~!l
-n.nts3
o.0276
0}._0~:-i
_____
-0.0661
11
l
-o.o5z8
0
:~=3-~-~.oo--oo o_·~_•j
-0.0853
-------------------.. n.o265
n.nTo;z
_1.0000 __Q.Ct,}_ll
-1
-O.O'il7
o.ot~o5
,.,. . _:: : : I
-0.1115
-----~-------------
0.0485.
-o.o .. n
0.0202
0.0276
o.oo~ll
:-_0,00~8
0.10'!1>
.
:-.<!...9~1!L...::Q .CI>.!!L.. ~
-- _____ j
26
varimax solution presented by Harman also does not result
in completelY orthogonal factors.
Identical Scores and Rotation
i
1
The case of rotation has been presented above when
I
iboth sets of factor loadings were rotated.
In the present
1
!case only one of the. two sets of factor scores is based
'
'!
I
;upon a rotated matrix of factor loadings.
Therefore, this
'
case reflects the correspondence between one set of factor
,scores based on the original principal components factor
.:loadings and the other set of factor scores based on a
varimax rotation of the original principal components.
The
icriterion established for this case is that an accurate
i
.[measure of invariance should reflect the fact that the
~actor
scores are based upon
'loadings.
t~
different sets of factor
Since the only change introduced into this case
'was that of rotating the second set of loadings, the
criterion is that an adequate measure of invariance should
!reflect the amount the second set of loadings have been .
rotated, i.e., T.
1
i
i
iResul ts for Each Measure of Invari·ance
l.
'i
- - -
- - - -
The coefficient of invariance and the coefficient of
I:congruence
resulte.d in a transformation :{tlatrix, Table 2,
!which will produce the varimax loadings if it is premulti1
!plied by the original matrix of principal components factor
:loadings.
The modified coefficient of invariance, Table 3, is
27
I
f--- · ---~~- ----~-- --
TABLE
1
_:i_~-
IDENTICAL SCORES AND ROTATION
COEFFICIENT OF CONGRUENCE AND COEFFICIENT
; t·
I
FACTOR
2
FACTOR
'- · ·~----~-~-
~F __ J:~A_i!~~~~---···----·~-------~
_,
'
~
1 ORIGINAL LIJAIHNGS
fliiiGINAL SCI:!RES 127ft X HI
-0.5518- -o.5138 -o.3'J63
-o.111o -o-.1H7 --~--- -Q.zsn -o.zft39 -o.oJ99 -o.Z612
o.orn
-0.1605
o.ofto6 -o.o397 -0.1942 _ ---~~~~--~-----~----~--~~-~-------~~~~-~~-~~~~~~--~~-~---~-~~-
2 OII~GINAL LOADINGS
ORIGINAL SCCRES 1274 X Jftl
(',09(11
o.oo5" -o.51oo
o.4583
o.2roo -o~o76a--~--o~-o<i3T_-~--o::3'.-91i-=o:-296i--o:z~s5·i-:
____:_(I._02I5 --~<1• ;30}_o~ _ _g_._;>_1~o___o_. 0469
f4CTOR
J ORIGINAL LOADINGS
ORIG1 NAL SCCRES I 274 )\ _14_1__ ___ ~-~- ---~----- ---~----~-~-~~-~~~~~~~---~~~~-----~~~-~-~~-~
0.3331
0.0279 -0.3492 -0.4563 -0.1969 .
0.1536 -0.26}7
0.3572
0.0589 -0.3169
_0.2857
0.2946 . -0.1212 __ -0.1078.
----··-- ---------------~~---~-~~~~~~-~~-~-~~~--~~---~~-------~---
~~--_f_~t:J!)IL_ 4 0111 G I NAL LOADINGS
!JR I_G PiAL _SC CIIE_S I 27 '!~JLl-"""-~=-:-;;:;;;---::--:-:-:-:---=-==::---::,-;=-:::---:::-==-'
o.499Z · · :.o.o,·zz - -o.c~~.,,
o.13M -0.5806
-0.2623 -O.llt44 -o.o219 -o.ZH9
c.12os
-O.llt83
FACTOR
-0.1656
~.3935
.-0.1455
ORIGINAl SCORES IZH X 1'•'---~-~--~--~-----~--~~~~~-~--~~~~~----------~-----~~
0.0070
0.4098
-'().2302
-O.•H29 -0.48Z7
0.0160
Ool09ft -O.ft787
5 OPIGIMl lOADINGS.
-0.~421t
0.1163.
o .2_}6 L~Q_,_Q~84_Q.,_Q8l)_!L__O. 0992 ____ -----------------------------~
6
6 ORIGINAl lOADINGS
ORIGINAl SCORES 1274 )(·H)__ -----------~--~~-~-~~-~~~~~~~~~~~~~~-~~~-----~~-~~~0.2~19
-0.3819
0.0969
0.1464
0.1146
-0.2909
0.4162
0.3449
0.0983 -0.2593 .
-0.2566 __ -0.0550 :-:O.J9_l_l_ -0.2347
·--------~-------- ·----~----------------~-------~--~--------
FACTIJR
_7___ _F.ti~TOII . 1 01\_I_GI !'4AL liJ.tiO_I 'IGS __ Dill!; I N.tll SC CRES 121_4__lL~-''--:::-::c::-:7""-::-c::-:-c:-::---=--:-::=--:--:-:-:::---:::-::=-::--l
-0.4169
o.111o
o.3914
o.Z463 -o.3613
o.2264
o.0462
o.4071t
o.01ft9 -o.2Z82
_ -o.1o8o
o.H19 -0.1890 _
o.n·n _
28
--···-·-------------------------·--·------------I-D·E--NT·--1-CAL
___ :·S·::::.~=~-=-~::~::==.~~~~~~~~~~~~=~~~~~~=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~----------------------~
~
"-.:..-4 ............4 .......... _ ... ________
-- -- -·
...u'tU
_____ _..:, ___ -·- ___ .eJDI.F.l.E.O...COEFJ"ICI£NT np IWapt•NCE
llnl--~.
,,!_·-·_ _._______
-(i~3:J~·o-:--z,;r·..-- :.:r~-.t;,;q--::-o:o·957 . -o.o91)8
-----~0.0771
0.0192. -C.Ol!I_B ... :-_0.0908 ....... _ ..
'•
·---~---· ........ · ·- ·• o.o7Rl
o.ool9
-o. JCJZ6
-o.tuto
-o.1z15
-o.o19a
-o.u97
o.cu6
------'-···------------!·-·-·······-------------·-------------<
o.-35-,~»
-----------~-~o~·;~o~•-~!____O~_zo~--_9~l~~~---Q~9~l"~-----------------------~-------------------------------1
i----1............
.1--- ..
0.)156
Oe2l'll'i
·0.0221 -r.2?11 -o.JA4~
Oo 2166 . _-0. CIICJ8 ...:-0.0 78.6 __ ........ --------------------------··--·-·-·-··--------------------·--·-·-----
fj
-o:.-
o. 0149 -0.1215
t-.1nto'
'l.07Bit
1'1 CJO ..
·-------- , .... ~~-'-- -:o:Oo 313 7__ .:~Oo1l39 __ :-0oll!11t .. _.. Oo 5821._. ________ --------0.1~1>1)
.0.61"-'0
1
-0.1011
OoC696
C.Z221t
-1),0022
-0.1196
0.2910
-0.1793
~o... ____---o:-2~16--o-:-5538 ·· .:.c:36-j7-=o~~v~io--=.,-~.,67 ·
r
;
. . -Oolt296
-1).2299 .. 0.1121
. .
..ll--.:··----. --.;~-iiao·-- -0. 0 i 56"".
-Ool't7C
---'~---------------·
•·
0.5931
---·---------. o.oq""
-'l.152~
. :..0.1179
0.2566 .
.
-O.It5'o8
0.276C.
-0.371o8
0.27!12
0.3016
o.oo5.r,
OoC899
-0.1362
.
I
c: iii-.-- ...:.a ~35iz- ---~: )iii;i,- ----------::()~)',;5-i ···.:.a:z;i.(;""'.::o:iio"ii--o:-2. oo-"o:i)b;j
Col965
-O.OCJA)
----------------·.-0.1877
o.17B6
_Oo5651 ... :-0o0665 ... Oo50C.5
.
•
.
•.
I
-o. 365.3 --· ·· ------:.o ~-oia3_____o~-29o~---=-o:5o22--o:o<is, -o.292" j
---- ·--···· .......... ------------------------------------------- :
.~.-
Oo1,2llt
-'
... ·.
o~ 1~to1·-::o~·o72i
I
---,,-;i'iii6·-·-c:,-:a·9jT___ o:zo79l
29
not equivalent to the transformation matrix presented for
the coefficient of in-variance and congruence.
Indeed, the
set of modified coefficients is not an orthonormal
transfor~
mation matrix since the sum of squares for a row do not
!equal 1.00; sum of squares of unity would not be expected
i
!since the varimax loadings are not orthogonal.
I
I
\ :I
-
I
Permuted Rows
!
The criterion established for an adequate measure of
:invariance in the permuted row case was that the two sets
~f factor loadings should be shown to be identical regard'
jless of the number .of rows l?ermuted.
This follows from
I
!the fact that despite the number of rows permuted, the
. !correlation matrix and. the method of factoring remain
identical.
Therefore, an adequate measure of invariance
.
I
based on row permuted
/30% Row
z
should yield an identity matrix.
Permutat~on
~- Coefficient of congruence. Table
lres~lts of correlating the app~opriate
I
.
4 presents the
factor scores
!
computed for the coefficient· ·of congruence.
Table 4 shows
that in'no case did a factor of the original scores correlate 1.0 with a factor based on the 30%.permuted scores;
thus, this measure does not yield-an identity matrix.
highest
relationshi~
The
measured by_ the coefficient of
congruence was .78 for Factor XIII and XIII'.
1
I
between the original factor loadings and the factor loadings
.
!
Inspection
of Table 4 also shows that the relationship between the
30
[_ _____________________________________ - --- -----------:--------TABLE- 4------------------------------------------------ -----'--- ---------- ORIGINAL SCORES AND_ .PERMUTED . .SCORES__.,._ 30\ • .ROWIL--------------- ----------------------- ----;'
1------------ ---------.-~
COEFFICIENT._!lF__CONGR!lEN
'
FACJOII
2
FACJOR
FACTOR
,
5
.
- - - - - - - - __
OIIIGINAL SCORES 1211t X 1"1
1 ORIGINAL LOADINGS
0.012,
0.6805
0.0160 -0.0954 -0.0,57
-0.0379
0.0655 -0.0,78 -O.OHO.
-0.0256
-0.0166
-0.10"2
-0.021...
0.0"83
2 ORIGINAL LOADINGS
ORIGINAL SCORES 127, X 1" I
0.6324 -o.o28io -0.0396 -0.0161
0.0242
0.0124
-0.0,99 -0.068" -0.0202
0.0003
-0.0355
o.o32,
-O.Oit89
-0.0082
ORIGINAL SCORES (211t X 1" I
3 ORIGINAL LOADINGS
0.75H
0.0015 -0.0005
0.0065
0.0160 -0.0284
0.0121
0.0003
-0.0202 -0.0205
-0.0502
0.0717
0.0636
-o.o,n
" OIIIGINAL LOADINGS
ORIGINAL SCORES 1271t X 1.,1
-0.0951t -0.• 0396
0.0015
0.6513 -0.0652
:-0.0325
0.0381
0.0 .. 31 -0.0682
-0.0060
:-OoO'tO()
-0.0137
-0.0111
-0.0839
-o.o~t6Z
-0.0669
o.o~oJ
9a9t!9~
FACTOR
FACJOR
5 OIIIGINAL LOADINGS
OIIIGINAL SCORES 127, X 1.,1
-O.D161 -D.ooos -o.o652 -._ o.7282
o.oo~t
-o.o599 -o.os,s
-0.0219 -0.0798 -0.0590
---------------- -- ------ 6 ORIGINAL LOA!)INGS
Ofl!(;l~~!, S(:QII~S 1271t X ~'!!
F~HQII
g~g?t,~
!!:90~l 9:16}9 9:9Wt 9e9im
!le99M "'9•9H§
"'lhQHlJ
-o.o~t57
-0.0165
-~---
"'
7
r
8
!lo661j4
=6.6411'1
6.62611 "'ll.6H1
7 ORIGINAL LOADINGS
ORLGINAL SCORES 1211t X ... I
0.029,
-0.0166 -0.0355 -c.o5o2 -0.0400 -0.0599
-0.0345 -0.0441
0.0113
0.032"
0.6546
-0.0227
-0.0503
-0.0718
ORIGINAL SCORES 1211t X ... I
8 ORIGINAL LOADINGS
0.0324
0.0717 -0.0137 -0.0545
0.0239
-0.0199 -0.0805
0.0202
-0.022.7
0.66 .. 3
0.0012
0.1289
ORIGINAL StOIIES 1211t X 1"1
9 ORIGINAL LOADINGS
0.0636 -0.0111 -0.0462
0.0203
-0.0274 -'0.0489
0.0114 -0.0043 -0.0632 -0.0281o
-0.0503
Oo0012
0.7711
0.031o8
ORIGINAL SCORES 1211t X 11o I
FACTOR 10 ORIGINAl LOADINGS
0.0602
0.0483 -0.0082 -0.0437 -0.0839 -0.0669
0.066,
-0.0290
0.0236 -0.0218-
-0.0718
0.)289
o.o3to8
0.7151
FACJOR l l ORIGINAL lOADINGs'
ORIGINAL SCORES 1271o X 11ol
o.oo91t
-0.0256 -0.0499 -0.0202 -0.0060 -0.0165
0.731o4 -0.0386 -O.Oiol1 -0.0195
-o_.ollt5
Oo0375
O.Olllo
o._on6
ORIGINAL SCORES 1211t X llol
FACTOR 12 ORIGINAL LOADINGS
0.0655 -0.0684 -0.0205
0.0383 -0.0219
-0.0,01
0.7436
o.ololtl -O.OH't
-0.0386
-O.Oiolo1
-0.0199
-O.OO't3
-0.0218
ORIGINAl SCORES 1271o X 1"'1
FACTOR .13 ORIGINAl lOADINGS
0.0127
0.0431 -0.0798
-O.Oit78 -0.0202
0.0268
o.olt4l
0.7815
o.oo95
-0.0413
o.o121t
-0.0805
-0.0632
0.0661o
ORIGINAL SCORES 1271o X llol
FACTUR' 1'1 ORIGINAL LOADINGS
0.0003
-O.Oit70
-0.0397
0.0003 -0.0682 -0.0590
-0.0195 -0.0331t
o.oo11s
0.1381
0.0113
Oo0202
-0.0281o
-0.0290
FACTOR
FACfOII
:=.Q. 1042
0.0375
II
10
11
12
13
llo
fACTOR
·-
1
31
two sets of factor scores was the
high~st
when a factor was
compared with itself rather than some other factor of the
.second set.
Thus, in the matrix of coefficients of
!
'congruence the diagonal elements show the highest relation-
Is h.1.ps.
1
I
'
I
Coefficient of
invar~ance.
As mentioned above, the
, · jcoefficient of invariance has two solutions, depending
i
upon which· standard s.core matrix (Z or Z*) is used to
compute the factor scores.
In this case·both solutions
:satisfy the criterion established for an adequate measure
'of in variance.
Modified coefficient of invariance.
Table 5 presents
l
Jthe modified coefficient of invarianee for appropriate
·jfactor score matrices when 30% of the rows of the original
i
imatrix have been permuted.
The results in Table 5 indicate
fthat.the modified coefficient of invariance is not an
!identity matrix •. A comparison of Table 5 .with Table 1
!shows that the same· values were obtained when the second
set of factor scores were based on 30% row permutation
1
(Z*) as were obtained with identical scores and procedures
lbased on Yarimax rotation.
I
I.§_Q_% Row Permutation
I
Coeff~c~ent
of congruence.
Table 6 presents the
!results for the coefficient of congruence for 60% row
ii
;permutation.
Again, the coefficient of congruence does not'
produce an identity matrix; however, the principal diagonal
'·
-·
J
.--------------- ------ -- --- -TABLE--5------ -------- ------------------------------------------------------
--iI
. ORIGINAL SCORES .AND PERMUTED .SCORES. ~-- 30~. ROWS--------------------------------------_1
-------~MODIFL£0-COEFE~ENT~NVAARKI~~~~~CE~-------
FACTOR
'- r-,.
I ORI~INAI
PFRMUTEO SCORES 12j~ X 1~1
l'i _,_,._Q.lJ591_ _
-0.0149
-o.C989
ICAOINGS
f'._9t,.i'L_,-_C),_~.9_?f>___ _::(l_~l!!}_IL_::9_~9-~
-o.n747
o.0717
-----~-----,;iifnli----i-nliTi.I-N4_i
__L!:-.\ii-tNr:s_____PI'Rilur-Eii--sc-oile
-n.zon -o.I'H2 -o.ot>n
s--·(27t;_li_1~r---
-n.11~1
n.9~'l7
-0.0697
-O.I5'i7
-0.1193
o.r293
0.0121
-----
- --.------------------- -----------------------
---:::_o__.J._313 ___,._o.21~-"..--=.Q_.,_?o85
r-
0.9.359
0.111>1
-O.l5'o2
0.91>08
.
FA!:TOR
7
-0.0707
PERI'UTFC SCORES 1274 X
0.0231
FA!:Toq
R ORIGINAL LOAOINGS
-------~-1-"49
Q. o~1
o.1
R
,
-n.!ilh7
n.ossa
-0.0907
o.3o6L::-O~os~!L
-0.1744
-0.071>1>
-O.H10_
~.0902
~-1 -lFA_r_T_'l_R__I_!' ORIGINAL L CAn I~G S
127~
PER!1UTFn SCORES__I_2_7_1o_X __
-0.0789 ___0.03~3_ _(!.99~.,_0280
r~:nt.-11----..:;,-:;;-15T----o~Ci91-~---:·a~-o2oz-
LCAOINGS
__
0.0966
_0.1022
-0.0323
0.0119
0.1055
0.9860
i
:.
- -- - ·
~
----- -------- --- -- -------------. -·-- .. ----!
11ol
0.0948
0.0756_ -0.1903
0.0226'
Oo01Z9
'
-- ·
·
-o. o7_"'_9__-"_o_._O>:. --_;:;;~;
0.0629
-0.0132
o.033lt
- ·--------------- ------------------------- ---- FA!:TOR 14 ORir.J~AL I OAIJJNG<;
PEI<MUTEO SCCRES--C27~ X-1~1
-0.~977
-O.IRAA -O.Oh84 -1'1.2763
O.Olt30
-0.0423_ ::-0_.101_8
n.978l
o.otol8
o.n112 -o.ooo1
o.oo.r,a
--------. :---·----------·------------------------~----------- -----------
0.0938
''
-;:.;;~; 1
PER!'UTFO SCORES 17.71o X 1ltl
. __-:I!.J.'I 7t, _____ll.OJ 8_9 _____Q,Q~_5_L_::-_Q,0_2_8~--- _O._Q'o73
,, ----
i
it;;·---- .- --- -·-- ----·----------------------------!
-0.1595
------------------------------- ... -- -- ·-- -FACTOR 17 ORIGINAl. lOADINGS
PFRMlJTEO SCORES IZH X 1ltl
O.n~R9
-0.07h~
C.I9RC
n.o172
0.0100
-0.0328
0.9621'1
~a___ ~_o._ca7~
;
--------------------~------------------------ ...
!JqiGINAI
_ o.o1
-o.0910
FAr.TOR II ORIGINAl LOADINGS
PfR~UTED SCORES 127lt
n.o74R
~.nn2h
o.Oh'il
o.1976 -0.0081
-n.onn2 -n.oo111 -n.096S
0.9605
n.n7R6
-o:o121·1
X 1~1
-o. n.2.1ll__ __::_'l.Q.Z.O'!. ________ :-'l.0518
0.0295 __ 0.0860.,. Q.Ob97 .. -0~0Z08
-n.119\ -n.n;>R3
0.0111
o.11o56
FAr.TOR n
'
I
PERMUTED SCORFS
bRC
1~1
-o. 025o
:-_C._1lt0~
-0.1101 ·-0.02Cl -Oo1Cltlt
FAr.TOR 9 ORIGINAL LOADINGS
PFRMUTFD SCORES 1274 X 1~1
0.3!177
.-1'1.74!>1'>
..0.0143 ___-0.1!>74 ___-0.31>21 -0.0227
-0.1970
0.0477
0.1771
0.1150
11
-0.031~
·- ---·----------
7 ORI~INAI. LOAOINGS
-'l.t3~L __ ,._o._'l92l ___ .::o_._tn3 L.::O. El'tl____ o.991tlt
-0.0406
,
l. -----17
o.o6~5
FACTOR
\ORIGINAl. LrAOINGS
PHMUTEO SC:JRES 127~ X lltl
. _______ .,-O.!litJ_l ___,_(I._J.5_6!'_____()~~11-~fl. ____ Q_._l_'>(>_!;_ ___,._p_.l.O(f>8_~------- -0.104 7 _-0.0813 __,.!l,,_1912 __ ..C!!l<lt&l_____Q_•_C/~_C!?__
-n.n7h'i
n.n772
n.1486 -o.ot>C9
\
·L
-c.C525
-0.181~
_o._012'ii __ :-_0._1<;5lt__
33
·•
!
-- ---------------------- .···-- --------- ·------------------------------- -TABLB-6-----------c-------------------------------- -------------------------_j
ORIGINAL SCORES. AND PERMUTED SCORES-.. .---60i
.ROWS---------------------------------------------i
1 - - - - - - - - - - - - - - - - - - - - - - ' C O E F F - I C Z E N T OF CONCRU&NCi:
0~3~~rGI ~~~ O~~:DII'i~~ OZ4~R~~~~~~9~CC~~~Og~-\ X l't) ·o:o~6·9----o~(ijjj<j---.::o:os-iz·--:-o:-oii7_____o:-oi.-i5----j
f-ACTOR
1-------=-_o_._!;5J!~O.JQ15__:_o._o~:o_a_o~o._lQ'\._,.__ _ _ _ _ _
.Z
---
------------------------1
2 ORIGI ~AL LOADINGS
OIUGINAL SCCI<ES 121ft X HL____
--------------------·------------c------- ----------·
-O.O<;Ol
O. 3337 -0. CZ67 -0.0';80 -o.OH3
0.0627
Oo0386
0.0087 -0.0~39 -O.OU4
____ .. _o .o 39_l; ___ -c. 0012 ___::-_o. C634 _____o. 0885_________________ --------------------------CC------------------------------------------------j
F-ACTOR
3_fACTOR .. 3 OldGI hAL. LOADINGS... ORIGINAL. SCC~ES... I274 JL Hl..________
--0.0240 -0.0267
0.4908
0.0846 -o.0416
-0.0046 -0.0280
0.0625
___________________-o .o6~4 __ c. C056_ -o. 0341 _ -o. 024'1_____ -----------------------:·---c _____________________:
OoC547
-O.C194
----------------------------1
1
• 4 _____ FACTOR
4 ORIGI ~AL LOADINGS
ORIGINAL SCCRES 1214 X 141 ________ ------------------------------------------------------~
-0.0490 -0.0580
0.0846
0.4063 -0.0121
-0.0035
0.0064 -O.Q061 -0.0165 -0.1195
.__ _ _ _ ___,.o.c1u __ o. 0351 ____ o._C185"'_:o. 0 8 2 9 ' - - - - - - - - - - - - - - - - - - ' - - - - - - - - - - - - - -
5. _
FACTOR
____ _______ _
-,-.fl~fAGTOR
5 ORIGIIIAL LOADINGS
ORIGINAL SCCR.ES 1274 .. .X ~4L---------------------------------------------------------------i
:g:g;~~ -~: ~:!L_:_g: ~~l~ ___ :g:g~f~-----~:~~--~
0
____________
~-~~-~~-~---~~=~-~~-~---~~=~~~~----~~-~~-~~--=-~~~-~~--~
un.. JI.lU.;c.~~····•·<==~~·~c-y-·
- _ - , "- - -~~~
0.0469
0.0627 -0.0046 -0.0035
0.0175
0.4418 · 0.0001 -0.0159 -0.0349
Oo0584
_______________________--:o .oo8c; .. -~_o. 0861__ __
ooo9 _____ o. o 3Zc._ ----------------------------------------------------------------------------------- f1 Ollllil MI. UlADINiili
o.
. t --1 ---- FACTUR
1
. .
.
8
tliUiiiNAl fiGtlllliS
I~ lt~Al
7 OR I Gl I>AL LOADINGs - OR
SCCI<ES -I ZH ..
O.OJ89
0.0386 -0.0280
0.0064 -o.0782
0.0164
O.Cl?_5__ _=.Q,OO?~___:-_O.O()_U
fACTOR _ 8 OR IGIIIAl LOADINGS
ORI
ia NAl
X_~~j__________________________________ ------------------------------ I
l
0.0001
0.4113
0.0076 -0.0177 -0.0592
____ : · - - - - - - - - - - - - - - - - - - f
SCCIIES __(_27~ .. X .l'tJ_________ ------------------------------~---------------------;
.----------- -----~~;g~~~ _ _ _ ~g: g~~! ---~~: ~:~: _ --~t ~~!L_~~-~~-~~-----------~~-:~-~~-~----~:~~~-~----~=~~~---~~-~~~~-----~:-~-~---l
SCCRES_i27~_X HL_____0_--.::-:=-:=--;;-:=:-:--~==--:--=:=-
\ 9
FAI;T_QR_ OlORIGII\Al lOADINGS _ORIGINAl
__
,__,
-O.C777 -0.043<;
0.0547 -0.0165
0.0636
-0.0349 -0.0177
0.0306
Oo~59
Oo0420
;____________________ o_.()ssl ____::-c. C35t _ -0.1842 ____ o. o 133_____________________________________________________________________ ~-------------
__ lo_
r --
FACTOR lC ORIGihAl LOADINGS
ORIGINAL SCCIIES IZH.X .. HL _______ -----------------------------------------------------O.Ob1S -0.0414 .-0.0194 -0.1195
-o.0~29
O.OS84 -O.OS92
0.09';9
Oo0420
0.3464
FACTOR-~-;O::~~~:::O:::DIN::lH:RI:::::l
~------ _ :_____ --~~; ~:~:_L_::~;n; i ___~_g: g~~~ _-~g :_~!-~~ .
i
11
StCIIES 12H X lU
m- --------------------- ---------------------------------------l
-o .o1o1______
-o.
12
,FACTOR 12 OR.IGI
~Al
lOADINGS
ORIGINAl
SCC~ES
127't X 141
o~a~---~-:~-~~-~----~=~-~~-~--~=-~~~--~-~:_01 ~~----j
.
·-------=~:: ~~-~-~~:~~~-~--~:-~~~L--~~-=~~~-- ~~~~~~------------_-~:.._~~~---~ 0.1~~-~~:_.022 ~-=~-~035_~-~~~-~~-J
13
---
fACTOR 13 URIGir.Al LOADINGS
ORIGINAl SCCRES 1274 X 1<41
_ ___ _ _ _ _____ _ _
__ _____________ ----------- ______ _1
-o.o6ac -0.0634 -0.0341
o.0785 -o.o9ll·
· ·o.ooo9 .;.o.oo21 -o•o646 -0.1842
o.U49
o.on1
c. t59~_ _o_,.4·~~-L __(!.05Zl
FACTOR 14 ORIGir.Al LOADINGS
ORIGINAl S(;CRES 1274 X 141
-o.ao4l
o.o885 -o.ozto9 -o.oaz9 -o.o513
·o.0326--··.:.o.oou
o.oz&8
-o.o<472. _ o. 0066 _ _o. o52l
_0.46.25________________ _ ________
__________________________ __
_
I
34
of Table 6 ·still contains the highest relationships.
The
highest of these coefficients of congruence is .4959 for
the relation between Factor IZ and IX'.
A comparison of
i
Tables 4 and 6 shows that the coefficients of congruence
1
I
i
!are lower when 60% of the rows of
i
~ere
z
are permuted than they
when 30% of the rows of Z were permuted.
I
The range
bf coefficients of cbngruence is .3337 to .4959 for 60%
'· !I
:row permutation and .6324 to .7815 for 30% row permutatfon.
Coefficient of invariance.
Both solutions of the
'coefficient of invariance resulted in an identity matrix.
Modified coefficient of invariance.
The resutts
'
[obtained
using the modified-'coefficient of invariance for
!
'60% row permutation are identical to those found for 30%
I
·!row permutation, Table· 5; hence, an additional table is not
I
.
.
[presented in,this case.
Thus, the modified coefficient of
i
linvariance does not yield an identity matrix for this case.
90% Row Permutation
1-
Coefficient of congruence.
Table 7,
shows the
results of the 90% row permutation case for the coefficient
I
iof congruence.
As in the case of 30% and 60% row permuta-
ltion, the coefficient of congruence does not produce an
lidenti ty matrix.
·j
In contrast to Tables 4 and 6, Table 7 reveals that
Ithe highest relationships between the factors is not
I
In other words, Table 7
!always in the principal diagonal.
I
I
!
i
!indicates that Factor I and Factor I
I
are not the most
35
FACTOR
1 ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
-l'.o5J6- -o.o517
o.ol93
-0.0122 -a.02A7 ---·· ---·-o;lfa9···-·o;·o<><Jo···--o~oto1·-·;..·o~o•iz4·-·--c~·roY9 ___ _
t---'------.c.''•IJC'Z
0.0433 -0.0546 -0.0081.
2
FACTO~
2 ORIGIN~L LOADINGS
ORIGINAL SCORES 1274 X 141
--o.a51 1 · 1.1 o 1 'I -o. ot.o4 - ~o. o1n- ··.:.a ;o·lz9 --------- ·-o~·roo4·-·--o~·oot;T··:.·o~-oM6---;..o~·o·ia3· ··:.:o~·.:n;,···-o.o265
•l.0493 -o.0662
0.1269
3
FACTOR
---··f
3 ORIGINAL LOADINGS
o.OlQJ
-o.o6U4
--o~-
o.o6tt
2
o.OI26
-0.012 2
-o. o1n
o. o6 11
o. ·'b •~
-o .os24
Oo0293
-Oo0045
0.1l23 __~0~·~G~1~1~~---------~-----------------------------------
-o.oJ44
o.06tt4
o.0216
o.1021
1---4-- . fACTOR -O: O::;~;~~~-O:·:~-~;~::u!_'I~R I G~:~~- ~~~-~~-~--;~~~--~ ·;~-;--·---------·------------------- .... --5
fkCTOA
· --
5 ORIGINAL LOADINGS
6
O~IGINAL
LOADINGS
- · -o~olz?:··· o.on 1----
-o.es1~
--- ... ---- .. ·!
o~·oro9·- ···o~·oo-z4·-·--o~·one·---~
ORIGINAL SCORES 1274 X 141
~~;g;~: -g:~~~= _g:g~~: -~g;~~~:
FAqO~
!
ORIGI"'AL SCORES 1274 X 141
.._
o.o4o1
__
,
---~~:-~5:~~-:-~~:-~~~~-:-~~::~2:~~:::::o-:o3o~:::.~~~:o~5~-::'
ORIGINAL SCCAES 1274 X 141
1--6
I'ACtOI\
7
~
"
t IIRIGI'IA'L LOADINGS
ORlGINAL SCORES 1274 X HI
o.o&9o
o.oo63
o.o644
o.on1 ~o.on9 ·--- ---o.o2s1
o.oo53 -o.o474 -o.o145
.
o~o1z1·----o~T223-·-.;.o~o5o4·----o~-ol4l---;.-o~-osu··--·
FACTOR
8 ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
o.o 101 -o. oo96
D. 021r.
- o. o1o9 -- ·,;,o .0274- ---------~.-o~o1>+6··-,;;,·o~ uso~o·-·- ·o.-o908 -- ·o~·oo77·----o~-Ol67 ___ __
-0.0221
0.0418 -0.0910
0.0428
------------ --~--- ------ -·- ------ -- - - - ---- ------------------------------------------- --------------------------- ----------- -- ----·
9
FACTO~ q ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
-0.0924· -rr;-ut1RT
O.IJZJ
0.0024
0.0301
0.0328
0.0141
0.0011
0.1059 -0.0010
0.0055 -O.OOR~· -0.029A -0.0110
,.
( ----------------------. ---------. -· ----.---- - ------.
----------- ---------- ___ .. __ . ------------------------------------------- --------------j _____
i
~-------
10
FACTOR 1r ORIGINAL LOADINGS
ORIGI~AL scoRES 1274 x 141
o.1o19 -o.H74 "'-o.os34
o.o148 -ii.0350
·o.o34r·=-o~·o5u--· o.0167"-'-0~-.,o1o ·:-:o.-o569-o.oD3J
o.oc74,
0.0121
-0.0394
11
FACTOR 11 U~IGi~AL LOADINGS
DRIGI~AL SCORES 1274 X 141
'"'~:~!~~
o.oz65 -'o.oMo
o.oz93 ·c.o.ont-· -o.o16z·-1
1 08
:RI
j
1 --~~--- ~~~;~~ 12 IJR lui::: ~~:DIN::
1
".V493
O.OOA7
-o.o54&
-0.066Z
~----------~o~.1~H2
-o.oo~ti
-o;o792
o.ossJ
•
f--~~ --- ;~CTOR-::1
14
'
o~o2il1
--o.ozzt---
0.0053
0.0418
·;.:o~-0336-- :.-o~Oit74
---o~o9lO
:~-::~I:~-~~~-~--;;;~---~--~~-;·-----------------------------
O.C433
I
-O.OD45
0.0194
o.l323
0.0111
-o.o524
-0.0361
·I
o~ooss·--;;;o.oo:Jl
--
------------------- ,------- -- j
!::;Gl N:: 0~::~;-::: IG::::9:C~~~-~--;~-;~--~--~~;-------------------------------------004:R
-0.0085
C.C074
:
---- -- - -.
0.... 0- ,·2·1-
--1
-o~029tl.
,
fACTOR 14 ORIGINAL LOAOTNGS . ORIGINAL SCORES 1274 X 141
-o.o·oA1
o.IZ69
o.6329- r.or:1fl .:.cl;o-534- ---- --·o;o!l'93--·.:.-o;oi<.s
o~o4ia--·.::o.oi16
_:-:_0._()_1_11_ 0.0498
0.0771 . -~·1681__ -·- -- ----------·· ---------------------------
-O.Ol94
-·.
36
highly rela.ted even though the loadings for these two
factors are identical.
Factors V, VI, and XIV show that
·•
they are more highly related ±.o themselves (i.e., to
I
i
Factors V', VI', and XIV', respectively) than to other
i
ifactors of the second set.
However, as in the case of
I
.
ractor I, the highest relationships for the remaining ten
!factors of the two sets are in the off diagonal·.
,I
That is,
~he relati<;mships are not highest for factors of the two·
sets with identical loadings.
Coefficient
ot' invariance.
The results of both
solutio.ns for the coefficient of invariance were an identity
I
matrix.
Modifie-d coefficient of invariance.
·rodified coefficient of invariance for this case are
!identical to the results presented for 30% row permutation,
ITable S;
· 1
I
hence, an additional table is not presented for
~his ~ase.
It will be. recalled that this table is not an
lidenti ty matrix. ·
Surni'flary of Results for Row Permutations .
I
The results of permuting 30%, 60%, and finally 90% of
!the rows· of
z
to form Z* reveal that of the three measures
.l~nly ~he coef~icient
,1.dent1.ty matr1.x.
of invariance resulted in an
The modified coefficient of invariance
I
idid not result in an identity matrix although the same
I!results
i
The results of the
were obtained for 30%, 60%, and 90% row
i
!permutation. The coefficient of congruence did not yield
I
1
an·· identity matrix nor were the· results the same for
.1
I·I
37
p~rmutation;
any of the cases of row
as successively more
rows were permuted the magnitude of the coefficients of
.. i~congruence
decreased.
•
Consequently, only the coefficient
i
!of in variance showed results which completely conformed to
I
!the criterion establ-i.shed for an adequate measure of
I
iinvariance.
Both the coefficient of invariance and the
.!modified coefficient of invariance produced results which
\. I
!
:show :that permuting rows of
z
does not change the measure
I
'of the similarity of the two sets of factors.
However, the
modified coefficient of invariance does not show an
'!identity
matrix because neither of the sets of factor
·,
i
:loadings are orthogonal.
Permuted Columns
.i
The criteria for the case of permuting columns of
z
to
I
!form Z,* are that an adequate measure of invariance should
I
Ishow that the factors are not identically related; in
addition, increasing the number of permuted columns should
I
!decrease the relationship between the factors.
These
!criteria are based on the fact that permuting colunms of Z
1to form Z* introduces change in the correlation matrix R*
such that R*
~
R.
Therefore, L* based on R* is not
Jequal to L based. on R.
Consequently, an adequate measure
I
!
•
.
!of invariance should reflect the
I
!loadings.
I
.
inequal~ty
between the
Further, as the number of permuted columns
I
iincreases, L and L* will become more disparate; that is,
the. factors based· upon the
or~ginal
scores are becoming
less related to the factors based upon the column permuted
I
I
38
scores.
..
30% Column Pe·rm:u:ta·t·ion
The result of the
coefficient of congruenbe in this case was an identity
I
.
matrJ.x.
In other words, the coefficient of congruence
'
j
indicates that the two sets of factors are perfectly
.'!'
'related when the second set is based upon the
intracorrelation matrix R* obtained from Z*, which is
z
with 30% of its columns permuted.
Coefficients of invariance.
~he
Tables 8 and 9 present
two solutions for the coefficient of invariance.
-·-
They
:indic~te high relationships between the two sets of
i
:factors; however, the two sets of factor scores are not
•i
,
perfectly correlated.
A closer inspection of Tables 8 and
I
~ indicates that one matrix of coefficients of invariance
I
lis the transpose of the other. This relationship is shown
~thematically
!
in the summary of results for the column
•
ipermutatJ.on case.
Modified coefficient of invariance.
t
/this measure are presented in Table 10.
The results for
Examination of
lthis table reveals that the two sets of factor scores are
I
inot perfectly
related~
I
i
i60% Column Permutation
~-
Increasing the number of permuted columns results in
'greater differences between the two sets of factor loadings
. I
Consequently, an adequate measure of invariance should
39
0.0163
2
FACiiJR- 2 OR I Gfh-~C LOADINGS
ORI Gl NAl sC.cRES .., 271t
0.0246
0.9137
0.0092 -0.0053 ~0.0105
------- .---- ·-:.:o.0390
o.C451
c.oaa8
d.lci24
'~----,f···-FACTOR
.
·a
OR l Gl N-Al. lOADINGs·
L____
· 10
F4CTOR-
O:O::;
g:g~~~
__-:g: ~~!!
Ool2.l8 ___Oo C2J6 __ ,
X ·y4-l ... --·---· ------------------------------------------------ ·--------·
7
0.0279 __ _9_oDl;!~_ _ _ _ Q_.5827
136
GIN:: O:::OIN::
O.OitO~----
ORIGINAL- SCORES- -I 271t lll 1,-,---------------------------------------------- ------------- -
------~-:Q,_D~ ll__~O._C<,Oit ___ 0._()598_
~-- 9····
_
:R I ::_:_:: :CCRES -IZ7_4-lll
_g:g:~~
. ~:~~~~
-o.oo.r,L_
-0.0420
Oo5567
_;0.2683
0.01>12
~1.4·1- - ------_--·-·-·:::c~----- -------------------------~~- ·------ .. -____ ,
..... o.1o4t._ ___
o_,l_7()~ ___:o,?_~2L __Q.~5_!1_~ ___:_Q.UB!> --1
F4CTOR 10 ORIGI NU LOADINGS
ORIGINAL SCOPES 127/o lll lltl
.
·l
o .011 2 ____ 9.!9_129 __ ::_a_. _1_1)1tl __ :-_o. o 388 ___ :::o .151t2__________:()_.o_t!]_~---- 9_.11t7I ____ Q..ll_f!:;_~ ___:J!._l_Q~?_____Q•.1'~~~~---J
-0.0409
'!.1268
0.062/o
0.161t0
'
'
··ir--f'-4ftriFi~fi-ol!icTiil\C-.::oA:ofNGs··-o;;:-fi;fi:.i.\i-·scci!Ts·n:7;.x-T-"---------------------------------------'
. ii.
-0.0536
0.9056
-J.0095
-0.021t5
0.0789
0.0917
-0.0429
0.90io9
-0.131t2
-0.0763
0.0293
-0.0770
0.1561
Oo0561
-O.COitl
-O.Cio52
--.FACTOR-~ ~O~~~GI N~~ 0~~~01 N~~ 1 , 3 ~R :~~~~ 3 ~COR~~o!~llo -i4)~~~~6 ~~---~-~~-~~~~-----~~~~~-~---~-:~-~~-~---~~~~~~;·--j
I
lll
_;0.075/o
~AtTORili'JitiGJI'.ALloi\cHNGS--iJRIGI~AL
!··
o.0521t
0.10R1
·o.o532
o.10a1
-0.1620
-0.1101
o.i11t1
-stoileslifi;·x-l't·I----------------------
o.0499
-o.1019
o.oaso·
-0.2781t
-0.0398_
FACTOR 14 ODIGINAL lDAOINGS
ORIGINAL SCO~ES 1271t X 11tl
-0.0294
o. 0996 -o. 0148
o.oo23
0.1372
-0.0566 __ -:_9~.063_8_.
Oo021t2·-·::o. 0516 -0.0989
o. R505
L _____ _
0._135_8_
-0._0322
~.9_.01.4~ .. -~o.
o._C992 _
t;l7_3_ _Q_ol..!l29
40
.,
:__________________________________________________ .. _. ____ - -------------------------------------------------------------------------- -------------- ---~
,--------------------------- __ ----- _____ ----- __ ORIGINAL- SCORES AND PERMIJTED__5.CORES__.,__3ll\ __CO_LUMN$__ _________ -----------------~---- --
,_
--.--~FAl'Xillf-COR-fGI
0.9476
-0.053,1o
l
FACTOR
--·--
~--
NI\L i:!:iAOINGS ____PEilMUTED_S_CC-PE-S--Izn-x
0,0242
0.06~1
-0.0242
0.0096
.-0.0051
0.1083 -0.0291
1~1
2 OR IGI ML LOADINGS
PERMUTED SCORES 12n X -ltof
O.C'2R'o
o. 91 37 -o. 0010 -0.0046 -0.0509
__ j).1251 _____Q_._D_~_8
----0~ D575- - 0.1 OH - 0.1000
~.;:0,0098
-
o.o~05
-o.0089
1
0.0128
-3 OR IGJNAL LOADINGS
PERMUTED SCORES- 127't X HI-------------------------------------------------------1
3---- _FIICTOR 0,0717
0,0093
0.7397
0.1548
0.16H
0,15't5_-0.07,l3
_0,0599 __ O.Qit_f!~--~Oo1~1t~--0.07Aa
0,1439 -0.1101 -0.01'o7
-~..--FACTOR- 4 ORIGINAL-LOA-fliNGS
PERMUTED SCCRES (27~ X 1~1
.
-0,0260 -0,0052
0.1835
0.9320 -0.1191
~0.0820
-0,076t, -0,0529
Oo0498
0,0625
L
------1
5
FACTOR
H
0.0280
0 ~ 0~~!Gf~~~o~g:fliN~: 080 ~ER~~~~~ 1 ~coR~~a!f!'" x 0 • 0408_~()~~__3~~- --~~~~~~---~-~~~~-~~ --~0 .-15-~~--t
Hi
--- ~-------- · 0.0523 ··· ·-o.;o673
- 6
__ 0.0586_
o.oast
o.l37J
·
FACJOq
6 ORir.INAL LOADINGS
PFRMUTEO SCCHS 127~ X HI
_-0,0183
0,0093
0.1044 -0.0866 -0,014'>
0.4372
-?.0771 -0,1665 -C.2784 -0.0566
0.0322
__ 0.5827
0.1045 -0.0872
.
-------~
o. 868't
-O.O'o_1_8 _____0._1}_()6_____ o,_~~-'l'9 __ _
.,7--FAttoli- --,-·ORIGff.i~l: LOADINGS -- PERMUTED SCORES ii71t_x_l'tl
0.0160
0.1562
!I
9
FACTOR
FACTOR
O,OR25
-0.0408
-0,0216
-0.0398
0,0541
-0.0641
0.0679
-
0,1012
8 ORIGI~AL LOADINGS
PERMUTED. SCORES 127~ X-l'tl
0,04JA
0,0425 -0,04~9
0.1002 _ 0.0530
0,5931
Oo0207
o;os5e -----o~ 098S --- -0,1359- -o.o1H -- -- -------- - --------9 O~IGINAL LOADING$
'PERMUTED SCCRES (27't X l'tf
0,1211 -0,0013
0,086A
0.0492 -0,01>90
________ ()oll2'o
-0.0040
0,0857 -0.0322 -0.0176-
0.5567
-0.2223
0.0852
-·- ---------- ------------------------ Oo20113 -0,2682
Oob583 ::-_().1027
---i
--~
,-ro-FilC-tr!•Cfc'-.,RIGINA[-ioAofiliG"s" ~-J>ERMUTED ·sti1Resl27<t x 1'~~"•.------------------------:
I
o.o279 -o.o4es ·-0.16&5 -o.o477 -0.15~>3
_o_.o066
_o.1zsa ____ _o.o61_2 ____,.o._17&_6 ____()~_76_58___ ,
-0.0452 ___ -0.-1968
0,099t,
0,1027
11
li--
I
FAC rnRT!"-Oi(i (;"1-Nii"C (O~llfNG s
PER 'lUTED stciiE~iT27~-~~-T~"f--------------------------------------~--------------,-;
~0.1049 ~0,0386 Oo0502 -0,0627 Oo0l69
-Oo0877
Ool448
Oo036}
0,0228 --C,Q408
o.9G51> -0.0429
o,o5l3
o.o24~t
·
,FACTIJR 12 OR rr.l NAL LOADINGS
PERI!UTEO SCORES ___ CZ7'o--X--l~-~--------------------------------------------------------~--------f
0.~05ij 0.0451
0.0639 -0.0040
0.09~t9
-0.1iolt6 -0.02b't
0.0716 . Oo116to
Co121>8
-0.0245
0,9049 -0.1620 -0,0515
!
-n-- F-ACTOR-T1nRTGI..,AL
0,1711
_0,0916
LOADINGS·--~PERM(JTEO- scolifS-IZ~~X
0,1)891
-O~ll42
-0,1352
0.7l'o8
0.0921
-Oo0989
Ool49b
l'ol
-0,2267 _-O.O.H'o
FACTOR 14 ORIGINAL LOADINGS
PERMUTED SCORES 127'o x· 1'1
-O.'l6b3 - 0,102·0 -0,0256
0,1282
0.2172
-0.02'>1
0.0291 -0.0755 -0.1018,
0.8506
. - -
..~Oo0714
___O,l_3b?___,_o._06_0'o___ O,C62l
,_0.05_7__8 __ O.OQ_1_jj__
Oo161ol_
41
·•
I
~:::~~-~=~~::·--~----..3--0-i·.-=:=:~------------------------------------------------------------------------~
-------------------------------------- ·------------------ -------. - -- ---AN---D--.
~---------------------------------------- -------ORIGI.NAL SCORES
¥~~·-~
.-
-
r--------------- -- '::."C:."--..c~z:,-_-_~_..,~"''=""--=----.----'"·..,-lo\ODIFI.ED COEF.FICTENT OF TNllApiANCE
_ __.__fui\CJOR
I OI!IGIML I OADINGS
-~
PEI!MllTEJL.Sc.LR.ELl27.-LJLl!tl____
o.nf'A -o.zsM ·-o.o13s -0.1112
o.o9~2
-0.1101 -o.10511 -0.10111
0.02211
o.c5011
~----- ______________-: P.~ lJ>. O.'L ... il-~9.;I.5_Q _____Q._C.t, Z.~---=--0 ._Q 62 3____ -------- ____ -------- _______ ... ________ . _____ . _____ . _________________ ----------------·
;
_____2_____ F}\C_t:Q_I!____ z__Q~_tG.l~_4LW.Ml_li"'_GS_____ !'B!4.VH !LS<;!l_PE~ __(2Jit. _X H.L. ___ .
-0.25hl
0.8251 -0.14A7 -0.1015 -0.1489
-0.0~93
-0.1164
0.0624
0.0229 -0~0026
~0.0416
-0.0655
-0.1395
0.0200
j __ 3. FACTOR...l OO,JGJ.ML.LOAOlNGS ____ l'E!'.1!UTEO ..SC:CIIES 127't.X.l'tL
. ________ -------·----l------·----------~g:g~~L-~g.:_~~;1 ___-,:g:~-~~! ___,~:M!_~~---~~:-~~-~~--~----···--~-·0'1~9 __ -0.15~5----~:-~-~~---~o--~~~~--~~-~~~~~-~.ACIOB
I
·
r·---·- ----------
t
5
I
_
6
-n.1o13
-o.1o5~
_... 0.111 L.
-0.1071
0.021Q
I__ u___
I OADJ"'GS
-O.C217
0.5117
o.CJ96
-O.CH8
O.l2l't
c.cna
PfRJWI£JJ....SCOIUoS-1216 . .X.l<!tJ ____ ------------
-0.0416 -0.1575 ~o.oOR6 -o.oq19
~o. O'tl.Q ____.,._c. 028Q ___ .,._O.J.11t2 _________________ .
-0.0654
·
0.1001
-n. 06)L___tl.J ?06
-0.0216
.
o_.762~
·i .
O.C252
0.0031
0.5117
-O.Oto81
0.4't98
-0.2036
0.0397
0.1216
-0.2035
0.5015
-0.0479
-0.1392 -0.1025
0.101~
-0.1201
... 0..1329. ___ c. 0251-.. -.o. 0412 ..
i--lO ..... ..fACIDIL11Lll.Q !GINA!
l
j
FACTOR. 9 QqJGINAL LOADINGS_ .PERI'UTEC SCORES 127ft X .1'ol .
,-,0.~46!1
,
-o~ 1 (cjj-·..:c,·;.1568 .. 1'
FACTOR. 8 Of!IGJ.NAL. LOADINGS __ PERI'UTED SCCRES 1271t X HI.
·
9
o.o253
0.1035 -0.1121
_. ----·-- .. ·---- _-----------· ... ,
-g·
6 ORIGII\'ll LOADINGS ... PERHUTH SCORES 1274 X llol
-~.1101
-0.04Q3
0.0969 -0.0214 -0.0373
0.3363
.-o .1 01 •L __-.o. n 73 ___ ,. c.18D3 .. ;-a •.o7.Z4t________ __ _.. __ .
~------0 . .0305
I
-o.ooa8
FACTOR
'
.I
PERMlll£.lL.Sl:CJlE..S_(2H_JL.H.L___.._ ---.. ---------··-:-:~---:--:-:--:---:--:-:-:-:-
0.1111
5 OPIGIML lllADINGS .... PERMUTED SCCRES (274 X 1H
~!j~ g·:::~ -g·~!~! o.8058
-o._on~__:>_-_o~~~~29
o:g;~;
r---:l__.fACIO.R...J_Ql!.JGUAI
8
I 0601 NGS
o.eB04 -0.2~01
-0.02l't
--~O.OCJ!L •• ::-.0. 011.6.c.,-.0 • .08..4L----O• Ol.'t6..._:_----------· ----·-· ... . .. . ..
FACTOP.
1
I
4 {JB I G I Ml
-0.1111
.
0.0502
0.0145
I OAD,NGS
PERH!!TEC SCCR£S-1.27~--V·l-------· -----·-- ----··---·
C.r\202 -0.049b -0.1326 -0.1567
C.16bL .. 0.002.4 .... 0.1020 ..
-0.03't8
0.0736
F.AC.tOB .. .l L.OR IG.I.NAL.LDAOINGS ____ P..ERHUI£Jl_ SCDRES..-1.27!o..lLl'tJ_____ . _. -----
-0.1612
0.~306
-0.1160
-0.0832
0.0471
0.0817
-0.007~
-0.0222
-0.1009
0.1709
0.0165
C.6519
---- ------- ----------------- -------------1
o. 0308
-o.onz
·o.o1~5
0.0015
i
---··- ·---------·-·--··---····-·--·-------------------ii
-0.0~10
-0.0631
··.·-- 0.1327
------- ·- 0.16bl
--· .......i
0.1206
0.0252
0.0023
-O.O't09
0.1021
42
reflect
factors.
th~
increasing disparity between the two sets of
That.is, the relationship between the original
factors and the factors based on 60% permuted
z
should
decrease from the previous results of the 30% permutation
j
case.
I
Coefficient of congruence.
This measure resulted in
, ·r-n identity matrix.
Coefficient of invariance.
Tables 11 and 12 present
the results for both solutions of the coefficient of
invariance.
matrix.
In neither case are the results an identity
A comparison of Tabl5ll and·l2 with Tables 8 and
'9 presented earlier for the -.. 30% column permutation case,
'
~hows
.
that ~he relationships between the factors are lower
. ''
when more columns of
I
z
are permuted.
As was true in the
Iprece d'1.ng case, one solution is the transpose of the other.
I
Modified coefficient of invariance.
I
.
bossible to
comp~te
It was not
t.he empirical results for the modified
1
,coefficient
of invariance in this case. 2
However, other
/evidence indicates that in this case results for the
I
.
/modified coefficient should be consistent with those
!expected for an adequate measure of in variance.
This
levidence consists of results for the case of identical.
I
!scores and procedures, the case of 30% column permutation,
l
.
.
lthe cases of 30% and 100% row.and column permutation
j··
1-----
~~2
These results were unavailable due tQ a complete
!changeover in operating systems at Western Data Processing
'Center.
~:::::= =:::::::::: -=- ::: --- ~;;~;~;:,:•:;,;;,;,-:.;;;~=::.;;:.::-·;;~-:;:;:::::::=:=:::::=:::==~~==::: j
~---..-----
__ .
.
_
. _____ COEF.FICIENT_OE:_IIDIARTANCE (Z)-
.
______
-·-----------------,
i27~-~~-r~r--------------------~
:----1--ncfnit--T-o!ifGftiAi. L.oAoiNGs ____ oRIGINAL scoRE:s
_
. o.q478 -!1.1055 0.2774 0.2808 0.0557
D.0653 -D.061t7
D.1420 .-0.0983
·1
-O.OZ41t -_-0.141t6
·
·z--· --FACtmr- 2 ORIGINAL LOADINGS
ORIGINAL SCORES 1271t X llol
o.03RO__
o.HOD
D.2134
0.2595 -D.0819
D.2765
--o·~-2634---..;·!1~1523-- :..D.1393
o.1121 --- ·--
l
,.
i
3--
FACTOR
l ORIGI "lU LllADINGS
ORIGI'IAL SCORES 1271t
_g:~j~~ :.6:~!!~ _g:~;!~ _g:~~~~
D.1575
--~;--FA'CfoiC-
4-·uifiGfNAL LOAO-lNGS
ORIGINAL SCOR{S--IZ71t
0.1522
0.0090.
0.0974
0.3702
0.0207
-O.HOD
0.1373
D.llll
0.1787
0.0612 ___ 0.2182_ ____ il.JO_Q_!_ ____Q~!~L....,
-j-i
X-1~,--·x·
L
D.o1~6---o~:-3~: ___-_o.D~~--~~D:_21D~~~D!_06:--
__
1~ 1
o:o211
D.3526
5
FACTOR
o; ORIGI'ML LOADINGS
ORIGINAL SCORES IZ71t X 141
-D.03t.7 __ :-0.156~
D.3740 -D.D2DO
0.5993
0.1330
-----------D-1208
0.1121 -- 0.2232-"' 0.0407
0.4131
h ORIGINAL LOADINGS
ORIGINAL SCORES I 274 X 141
-0.0624
0.2913
0.1148
0.0856
0.1367
0.2785
-O.D773 -0.1722 -0.1214
o.06D6
-0.1614
.- l
- -
6 -_---FACTOR
-
.
FAc'i.-n_R"_7_':J<i!GfN-~L-I..-oAniNGS
1
-o.o1~>5
0.1001
-~~-
FACH1R
o.D2~4
-0.0314
oRIGINAL scliR~s-- 1274
o.i577
o.312D · D.3lt76
-0.0362
0.1682
0.21t20 ·__ 0.1891. ____0.1395 __
.
-0.0786
D.6D36
0.103'1t
0.1258-
-0.230D.
0.0567
x·r,;,----- ------------------'--------~
_
-0.1'160
_D.21t22 _ -O.D655 ____ 0a222f> ___ O._C895 ___
1
~ORIGINAL LOADINGS
0.1195
-D.1260
0.0329.
<f.1263
ORIGINAL SCORES 1274 X -141
0.0943 -0.1515
0.1459
0.5618_
C.197l
o.D992
9 O~IGJNAL LOADINGS
ORIGINAL SCORES 1274 X 141
0.0742
~.0268
0.1064
D.2695 -o.t815
0.1230
-o.oo5o
·o.to67 -0.0639 ·-o.o588
·- --- · - · - - FACTOR 10 ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
0.0645
0.1888'
0.0042 -0.2787 -0.048D
-0.0891
-0.1~51
0.2847
0.0797
0.1657
-D.2D39
FACTOR
'
10
,_
-
,11
i
. --I
-----o--FATfnJi"f3--fJR)Gfr-iAli:oADTNGS-ORrGTNAL SCORES 1274
0.1R&8 -C.1Zl9 -0.1749
0.0085
0.1897
-D.D028 -0.2025
0.4246 -0.09D3
~
-0.24D8
o.0729
-o.DD05
D,t,9t,3_ -C.1804 __
-D.l621t_
fACTOR 11 ORIGfNAL'-LnAOINGS
ORIGINAL SCORES--i27~--x--il;i_______________
--- -------------~---o.ol3?
~. 2054
-D. 0556 -o.o798
o .oao2
-D.1079
0.1204 -0.0885 -D.1124
0.6922
Oo01~2 -0.0032
D.Oit57
----tz-·---f.--.l\i:fijli-Y2--oi\"'GTi·iAL LOADINGS
ORIGINAL SCORES 1274 X 141
0.0~56
-0.1127
0.1732
D.D44~
0.0812
-0.1600
0.0279 - D.8D65 -D.17~7- -D.DD72
. l't
D.2060
. FACTOR. 1.4. ORIGINAL LOADINGS
ORIGINAL SCORES 1274
-0.0977
D.1525
o. 0291
0.1291 -D.Ol72'
- --0~1)68.4 -'J.016t, -D.12760.7257
---------
----- --------0.1D93
0.0997_
0.599_4
----------------O.ll46
r·
----------;,'
-0.1251 '
().3209
'i
~
x--1'-,.-,-------------- --- - -- -- -------- ---------- -- ---------<!
-0.1028
-D.0339
D.1608
o.D374
0.0996
c.12o6
-D.0179
0.0815
X 141
0.0418
44
..
I
TABLE 12 '
, .
j'
------ -- .. - --.----- ··- -·-··-- -.-------------------- --- -- ------------------------~--ORIGINAL SCORES AND PERMUTED SCORES ~ _60' _COLUMNI?___________________________________________ J
·--------------------------------------------------------------.-----
.
.
i---------------------------------------------, ......... --- . .. .. .
'
· - - - - - - - · ---------,---- <:;OE_FFI<;_I_ENT__Q_F'_I_NVARIAN<;ILJ.Z.!)
-.-1--F-.cTnil----r-nlfi GiNAL'(IiAifiNG·s-- PERMUTED SCORES- 127~-~~--11;1----·-0.8478
o.D378
0._1398
0.1524. -O.D367
-0.0626
-b~0141 ..
0.0057
0.1870 -0.0977
i_
7.
)-- 3
F•CTOR. --2
ORIGIN~l
-:~~::t, :: m: ~, :: :;::
IGIN4l LOAOINCS
0.2~06
~0.07'18
5
I'ERMUTED SCORES
127~
::-:m "'· ....
u
FACTO!\ .. 3 ORIGIN.L LOADINGS.
PERMUTED SCCRES 127~
0.2174
0.2733
O.l'l'l3
0.0'173
0.37~0
-0.0556
0.1732 -0.1749
0.0291
--,.--f4CT0r:r-4m~
I,..
LOADINGS
0.2594
0.0444
PERMUTED. SCORES -(27.,_
0.2114
O.OOR5
0.3102·
0.1292
I ;,--- ·
f
0.1H9
- --0.1577
-
:Oo!)ll57
0.3122
-0.15H
X
0.2785
-o. H&o
·x
-o.o1ao
i
_
I
~--5~~8---'~~~~---~::_89_1_
11tl
ORIGIN~L0.1010
LOADINGS
PERMUTED SCORES 127~ X-l~)
0.2109
0.1035 -o.2261t
- o.on2
-:0.2787
1~1
-o. 0655
o.ll48
. 0.26'13
-0.14~~----~~~1_81~~~:Q_~81=-~
0.3~75
fACTOR. 8 ORIGJNALLOAOINGS
PERMUTED SCORES 1274 X HI.
'
0.2423
0.2180 -0.0445 .-().0786
0.1404
0.1>036
-0.0887
0.09'11>
0.11>08
0.1206
L
- --____ 0._00~2_
'
o.2422
'l
0.1886
·-- ·-0.1122
-Oo106~
Oo09H
X-f...-----·-- --·-------· ---~.-------
-0.1611t
ACTOR
-----,
'·,.." .'·"" -~ '· '"'- ~ ~' ·'"'---'" '"]
X..1"'t.
5 !lRIGI NAL LOADINGS
PERMUTED SCORES 1274 X HJ.
0.0557 -0.0820
0.1574
0.0206
0.5993
0.1366 --·--o.oai>'7.·--r,~ 0811.
o.18'l8 -o.onz
-7"-fACTnii" 7 OR fGI NAL LOADINGS
PERMUTED SCORES 1274
-0.1447
C.Ob13
0.1313
0.3525
0.~132
-0.1204 ·:..o~-10'12 -o.033'l
o.1'l58
·
11
------------- ----.. -----------
1~1
-o.01'l'l
FACTOR. b ORIG;NU LOADINGS
PFRMUTEO SCORES 12H
-o .0242 o. 27&5 o. ou5 o. 0216 o.1n1
-0.1'l81 -0.11>00 -0.1028
0.0419
10
o._H91
FACTO'!
·r-·--..
x·
-o.o363__
0.1259
-0.2038
o.2059_ .. o.ono ___ .
_0.291>~ __ :-0._2~08
__-:_0._000~----
.,
0.2228
. -0.3~5 .
0.4943
-0.1621
'
'
fACTOR 10 ORIGI NIILTOAilTiiiGS_'_PERMUTED SCDRES-Th~-Xl~-------,-------0.13'18
0.1531 '-0.01>46 -0.2300 -0.1531>
0.0568
0.0895 -0.0100
-0.1250
0.320'1
0.0998
0.0834
- ---····- ----- ...
----,:-icn)R-TCnliTGHiALTblioJNGS
PERMUTED SCORES 127~ x HI
0.01>52
·0.21>35 -0.0795 -0.1700
0.120'1
~Q-0772
0.1001 -0.1258
0 .1>'122
O. 027'1 -0. C0_28
O. 01>86
-0.1805
0~59~_
---~--------------------------
ORfGINAL ioAOINGS. · ..Pe'RMUTED scoRES -i211t x 1~1
-o.~&47
-0.1522. -o.0461
.o.un
o.1123
-0.1121
0.0153 . O. 801>5 -0.2025 -0.011>1
-0.0051
-0.1852
-\
tz·--;f'actnR--fZ
;
t . .. .
..
~AtTO~f3-0R-l<:-t-NAl-LOAOi NGS
0.126~
..
0.1Q65
0.281t7_
---~~--ittf---------· ... -------- --------------------~
-0.121~
-0.0362
o.&'l70
-0~0638
. FACTIJII. i4' DR IGI NU .LOADINGS
PERMUTED SCORES 1271t X ~~.-o.~'lK~
0.111 'l -0.0222
o.l78&
o.o~ooe
o.0606
-- o.o455 --.:a. oon · -o.o9o2
o. 7258
· · -·
0.1683
0.0992
-1).058'!' ____0.16.51 ___ __
0.1418
-0.0032
lit
- PERMUTED SCORES 1274
-0.2747
0.1111
0.2232
0.42.46 -o.1271
-o.on5
1--- ------.----·
-0.13'12
·-0~1787
0.0795 __
45
presented in a later section of this paper, and additional
interpretation of equations presented in the summary of
results for this
·•
s~ction.
!
i90% Column Permu:tat·ion
i-
I.
1
~he
results expected for an adequate measure of
.l1nvar1ance in this case are similar to the results of the
\,.
I
[30% and 60% column permutation cases, i.e., the
relationships between the two sets of factors should not
be an identity matrix.
In addition, since more columns are
permuted in this case, the other criterion for an adequate
measure of invariance is that
the relationships between the
-,
factors will be even lower than they were for 30% and 60%
column permutation.
Coefficient of congruence.
The result for this
measure is an identity matrix.
Coefficient of invariance.
Tables 13 and 14 show
that the coefficient-of invariance does not result in an
identity matrix.
Tables 13 and 14, when compared with ..
Tables 11 and 12 of the 60% column permutation case and
Tables 8 and 9 of the 30% column permutation case, show
that the similarity between the two sets of factors is
lowest when 90% of the columns of Z are permuted.
Tables 13 and
14 indicate that in this case RIVz
i.e., one matrix is the transpose of the other.
Again,
=
These
tables show that·only Factors I, X, XII, and XIV remain
most h~ghly related to Factors I', X', XIIi, and XIV',
i
i
46
.~::::::::=~-=~:::::::~:::=::::~;~~;~=;:::::..::::~:: ;.:;;..;,; ;,;.-; :: ::-::::
--·i-----f-AC'TOR___l -tiRIGINAl LOADINGS
0.6726
-0.1l3~
0.0714
C.1A13
ORtGlNAL SCORES C271t
0.3414
-0.0286
. 0.3803
0.0436
X 1,,-·----------
C.0224
0.1081
l
. UCTOI!
2 U~IGIN~L LOADINGS
ORIGJ'iAL SCOI!ES 1274 X 141
0.1088
.
o.1ry72
o.1614
9.3109
o.2453 -~.1718
·-----·--·---'1.3013 -0.0790
.O.OitOO
·o.2139
I
3
-FACTO~
3 URIGIN~l LOADINGS
ORIGINAL SCORES 1274 X
0.1434
0.2785
0.09~4
0.1526
0.1415
-0.1412
Oo1753 -0.2137 -0.0363
~- ..~AtfffR---;,-·ORIGI
0.2~45
. Ool517
'"-o.~:
MlLOAOINGS
ORIGiNAL ·sCORES 1274 X 141
-0.080il
0.0644
0.0274 -0.008~ -0.0771
C.2945
0.1085
0.1070
::=:~-- -I·
--···------·------0.2309
Oo3639
I
0.0739
-~
0.1808
!
0.21188___ ().2525__ - ..0 .• 1092_ -- 0.20)6 ___ j
I
'"""_~':'"'_
'·". _:·"''J
0.3001
0.2180
-0.0598
0.2595
0.1422
. 0.1918
o.c19S
-0.1176
. 5. FIICTOR
5 UI!IGI t.:AL·LOAOI'IGS
IJRIGI"'AL ~CORES 1274 X 141
O.OCb5 -0.2204
C.4027 -0.1414
0.4563
-0.0042
---·-------··-1).2811
0.2915
O.l30b
0.0663
l
6
foACTO'I
6 IJRIIH NAL lOADINGS
ORIGINAL SCCRES 1274 X 141
0.1265
o.OH7B
o.D496
o.0346
-0.1932
0.1056 -0.0044
0.0833
o.o~o2
0.1233
0.1324
.·,l~ACTOR-lilRIGI~AL LOADINGS
-~.1~42
0.0007
If ... FACTOR
0.2910
O.Ol06
ORIGINAL SCO~ES-1274-X 141
0.1189
0.2739
0.3759
0.0699
-0.1HBO . 0.0890
-0.0628
0.0254
-- ---·-i
,
!
8 ORIGI'IAL LOADINGS
O~IGINAL SCCRES 1274 X 14i.
0.2682 -0.0065 -0.049~
-0.19~9
0.2671
____ o.~ 701 __~() .• _1()4 L-=O_!l g~_::(l_!l !;."_! ___~ !!~~-l
-0.1942
0.3367
0.0177--
,--~-0.0422
1
I
9
·-fACHIR q ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
0.2Z56
-o.o291
o.0183
o.2139
o.3751 -o.Z493
-0.1746
0.1888 -0.1054"
0.0369
0.2078
-0.0722
0.3725
I
:o-f.ACTOR-lO--fJR IGII'IAL LOADINGS --ORIGINAL-SCORES -1274-X--14t _ _ _ _ _ _ - 0.01189
c.Z776 -o.otz5 -o.1'1t.6 -o.o630
-o.oo11 -o.o5o9 . __ o_.tsi~t __ _:-0.2274
-- -0.261';
'l.ZZ46 -0.0674
0.1056
l
11-
12
.13
H
0.394.2
foACTrlR--l(ODIGI~AL
-0.1247
o.19Z6
LOADINGS-- ORIGINAL-SCORES 127lt-X--l4f- -- -----o.zo1o -o.oo65
o.l262
0.1843
0.1596
o.0353
o.21ss
o.1367
Oo0788
-0.1983
-0.2U!)
0.0997
0.2516
-o.o~43
.fACTOR 12 URIGI NAL LOADINGS
ORIGINAL SCORES (27lo X 14t
0.1552
0.0736 -0.1049
0.1233 -0.2570
0.2117 ___ 0.1151_. Oo2357
-0.01l9
Oo37lt5 --0.0991 -0.0582
- ---------------·---FACTIIR 13 ORIGIN4l LOADINGS
ORIGINAL SCCRES 1274 X 14t
0.2689
Oo0615
Oo0991 . -0.09blt
0.0752
Oo1127
-0.0203 -0.16<i3
o.2~~1
-0.1945 -0.1409 -0.0951
FACTIIR 1to ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
-o.o275o
o.1842 -o. 0354
o. 0623 -o .o765
·-o.oll9 . _.o_.1t,Z5.
-··0~1315
Co OliO -Oo0925
· Oo6565
-0.1123
o.0630 ._-_0.0168
O.C761
__j
47
"
..
TABLE .
~~ __ ----·-····················-······---------------·------~
········-· _ ..... _.ORIGINAL SCORES AND PERMUT_ED.• ~~9-~l!.•:-.. ~.I!~.S.Q!-P.I:I!!.!!••••••••• ___________________ ~---1
.[__ -==~--~~~--~---. ·- .
.
1
~IICHIP
.
·--~--~~~-~ COEFF~CIENT
OF
INVARIAN~~-(=)-------------··------···
I IJR 11'01 NU LOAiliNI;S
PERIOUHll SCORES ll71t X. lit)
~.&727
1J.1o21
o.14l,
o.Z547
o.oo~6
o.oaoz
-O.I21t7
Co12l4
0.0614 -0.0214
-O.I54l
2
fACIIJII 2 OQ JGII\;ll LllAOINGS
PEQI'UIED SCORES I 27ft X 141
····-··-···--· .. -- .0.(171 It
0.161 It
C. 27~6 · o.Oh45 -O.:t204
Oo1261t
o.2o1o -o.z57t
o.C991
o.IB4l
I
-0.0289
o.0885
0.2909_-0.00&4.
Oo0182 ..
0.11&9
-0.0495
Oo2139
-O.C125
0.2141
-o.ns8
o.nso
•0.1966.
Oo1919
-0.1564
Ool72S
-0.227)
o.ooe;a
Oo0422· -0.1746
•0.2~15
OoC205
Oo1942
Ool887
o.22<o6
-o.1&bO
0.3366
-o.1os~
-o.o~75
3 ORIGINAl. LOAIJI~GS
PtQ"UTEO SCQQES 127ft X 141
0.0273
0.40l7
Oo3411t
Oo310.8
Oo09S4
9o0878
. •··-- ·····-·---- _ ~~-~-!l61t
Oolll7 .,.0.0964 -o.0351t
~
fACTOR It ORIGI .. AL LOADINGS
PE~IOUTEO SCCQES 1274 X.141
Oo1801
Oo2452
0.15Z7 -0.0081> -0.1415
Oo0497
Ool26l
Oo1152
0.0752
OoC623
3.
o.Z&80
0.2777
u:crnR
5 IJRIGINAL LOADINGS
PERM~TEO SCO~ES 1274 X litl
OoD7.Z'o -0.111CJ
0.1415. -o.0111
0.4563
0.0}45
Oo1842
Oo2)56
Oo1127 -0.0765
9
fACTOR
9 ORIGINAL LOADINGS
PERMUTED SCORES 127ft X 141
0.0736
Ool093
0.7.976
IJ.21R1 -0.2542
Oo2596
•tl.l9112
·0.099'8 -0.1123 -:l.Ol69
• •0-I'ACT<tR..lD OR l·ta NAL liiAOING·S·
P~MIOUTED SCORES 1274 X 141
Oo1R11
1Jo2035 -0.0655 -0.05CJ~ -Do1l81
0.1422
-IJ.2H10
0.2516
0.0664
Oo0761
1 ..
I
lZ
fACfO'I 11 IJII I Ill N4L LOADINGS
Pf'I~UTEO SCORES 1274 X 1•1
•0.1231t
0.1074 -0.1411
~.1516
0.2912
Ooll2'\
Ool925 -O.OI1C
O.Z55b
Oo1316
f•Cfll~
12 ll~ IGI N4L L'lAOI'lGS
PF.~>~UTEO SCORES 1274. X H!
•0.0790
t1.17~2
~.zq~~
0.2917
ColOS6
0.3746 -0.194~
o.o110
0.1~11
-o.os~z
l J - -... Cfoq·-13
I
1
...
.
~~-
OP IGI NAL LOAIJI NGS
"E'I'IUTEIJ SCORES 1214 X HI
-o.o2e6
n.o4oo -O.l13s
o.1os4
0.210~
-o.oo•s
IJeZl89 •Oo0994 •Oo11t09 •Oo0924
__"A_Cr_n_~,
_
14 ONIGIN4l liiADI~GS
PF.IC~UTED SCCRES 121• X .l'ot
Oo0'\35
Oo213R •O.Ol~l
0.1010
D.0663
O.OIIU
0o1165 •C!o05"2 •0.0954
Co6566
..
O.Q~~O
_ j)._O.lll __ I)_.0_)7_l___ Q.._l9.~.6-
48
respectively; the remaining factors show highest
relationships in the off diagonals.
Modified coeffic'ient· of inva·ria:n:ce.
The same
.circumstances hold for the modified coefficient of
!invariance in this case, as were presented in the 60%
l
!column permutation case.
That is, the empirical results
I
.lare not available but may be inferred on the basis of
' Ii
results already presented as well as the relationships
described in the summary of results for this section.
:summary of ResUlts for Col'u:mn Perm:u:tations
For 30% column permutation the coefficient of
·"
'congruence was an identity matrix.
I
.
Increasing the number
I
:Of permuted columns to 60% and 90% did not change the
'i
!results.
I
In each of the three cases of permuted
colu~s
the
!coefficient of invariance did not result in an identity
~atrix. ·
The cOefficient of invariance did show that as
~additional
columns of Z were permuted the relationships
between the two sets of factors decreased.
~the
Consequently,
results prese.nted for the coefficient of invariance
~~onfo~
to
~he· results
expected for an adequate
11nvar1ance 1n the case of permuted columns.
i~esult
~a~u~e
of
An add1t1onal
of the coefficient of invariance was that for each
i
i
:of the three cases of column permutation the matrix of ·
I
coefficients of invariance based on Z was equivalent to the
transpose of the matrix of coefficients based on Z*.
The
49
=
following equations show that R ,
1 Vz
definition L*
'
jloadings.
!iP -1
. •
=
PL
R
By
IVz*
where L is the original set of factor
-P is a permutation matrix; hence, P
=
P1
=
Substituting
[7]
I!
'I
\.
I
:By substitution in .equation 3
;
[8]
R
IVz*
=
L-1
=
Ll
PL.
yields
I:Transposing
.
. RIV.z
I
I
R
r-v
I
.I
~hich
I
I
PL
-1'
z
Thus, R 1 = R
Z*
IVz
IVz*
Results for the modified coefficient of invariance in
is eqUal to RIV
!
i
the case of 30% column permutation did not reveal an
1
~identity
matrix . . These resultS, presented in Table 10,
'!indicate that a modified coef~icient is consistent with
the.criteria for an adequate measure of invariance in the
1
I
I
.
•
:case of column permutat1on.
I
Foefficient are. as
The equations for the modified
.
follo~s:
I
=
·,-I
I
=
1
So'
S*o
N
1
N
L
-1
z0
I
L*
I
I
I
I
=
L-1
-1'
zo
L*-1
I
I
I
I
Ii
50
L*-1'
=
Since L*
'
PL, by substitution,
I
I
i
l
=
I
I
Iiit 1.s
· ev1.'dent
I
.
p
f rom the last equation that as the
.
, jpermutatl.on matr1.x (P) becomes increasingly disparate from
'\
'!
!I,
L-1 an~ L*-1 also become increasingly different.
i
,consequently, their product departs further and further
i
I
from an identity matrix.
i
While, as noted earlier, it was
not possible to obtain the empiriqal
Fe~ults
for 60% and
90% column permutation, these considerations indicate that
this measure agrees with the criteria for an adequate
measure of invariance .·
Row and Column Permutation
The criterion for an adequate measure of invariance in'
this case was no~ an identity matrix.
I
In addition, it was I
expected that increasing the number of permuted rows and
columns should reduce the magnitude of the coefficients
presented for each measure of invariance.
The advantage of;
this case lies in the fact that by permuting both rows and
columns of
z
to form Z* more variation has been introduced
between the two sets of Z scores.
30% Row and Column Permutation
co·effi"cie·nt of congruence.
Table 15 ·presents the
results of correlating the appropriate sets of factor
l
'Il
51
TABLE 15
ORIGINAL SCORES AND PERMUTED SCORES_,.. 30\ ROHS . .AND. .cOI.llMHS--------------------------------COEFFICIENT OF .. C O N G R U = = - - - - - - - - - - - - - - - - - - \
FACTOR
i-
1 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X
0.6805
0.0122
0.0161 -0.0953. -0.0458
-0.0255
0.0655 -0.0~76 -0.0470
/
1~1
-0.0378
-0.0165
2
FACfOR
2 ORIGINAL LOADINGS
ORIGINAL--SCORES-127~-X--1~-;.-------.-------------- -------------------0.0126
o.6324 -o.o2o5 -o.0396 -o.o161
o.o242 -o.o355
0.012~
-0.0490 -o ooa1
20
3
FACJOR-O:O:::GI:::O:::DI:::0 :RIG::::O:CORES 1274 X.
I
0.0160 -o.o2a4
0.7533
o.oo15 -D.ooo5
o.o~6~--~D.0502
-0.0202 -0.0205
Co0128
Oo0002 . -· - --- ---------
1~
FACJOA
4 ORIGINAL LOADINGS
ORIGINAL SCORES 12H
-o.o395
o.ool5
o.6573 -o.0651
0.03~4
6.0430 -0.0681
·x
1~1
o.o111
o.0636
_o_~_-_-l-_l-:·--j-
-_---__O••
~
. - -----------0.013:J'. _-0.0112 _ -o.o8~0
-o.o325
-o.o~ooo
FACTOR 6 ORIGINAL LOADINGS
ORIGINAL SCORES 1271t.X -1~.-0.0379
o.0242
c.oo65 __ -o.0325
o_.o032
__ o.763D
0.0094 -0.0407
0.0268 .-0.0397
0.0294
0.0239
0.0202
__ 0.0602_·
FACTOR 7 ORIGINAL LOAOIPtGS
ORiGINAL SCORES-1274 X 1~1
-0.0167 -0.0356 -0.0502 -0.0399 -0.0599
-0.0345 -0.0441
0.0324
0.0172
0.6546
-0.0226
-0.0503
....... j
____________ _
0.0239 -0.0227
0.66~~
Oo0011
0.1289
-0.0502
0.0012
Oo7711 ..
o.031t9
-0.0779
0.1289
0.0348
o. 7151 .
-0.095~
-0.0061
I
!
!
6
7
_I
FACfOR
8 ORIGINAL LOADINGS
-o.104o
0.0374
9
10
11
12
I
..
I
FACTOR
o.o325
-0.0199
ORIGtNAC-stoRe-5--(i7~--X~--14t
c.o716
-0.0805
-0.0137
Oo0202
-o.0545
0.029~
...
9 ORIGINAL LOADINGS
ORIGINAL SCORES 12H X 141
-0.0277 -O.Oit88
0.0637 -0.0109 -- -0.0464
0.0204
0.0115 -0.001t2 -0.0632 -0.0286
FACfOR. 10 ORIGINAL lOADINGS
ORIGINAL SCORES 12H X
0.0484 -0.0081 , -O.Oit37 -0.0839 -0.0668
0.0216 -0.0218
0.0665 -0.0291
·~·
0.0602
FACTOR 11 ORIGINAL LOADINGS
oili(fiNAL -SCORES(21itx"Tto,l-------------------------------. -0.0257 -0.0498 -0.0202 -0.0060 -0.0165
0.0095 -0.03~5
0.0376
0.0113
.0.0236
0.7344 -0.0386 -0.0413 -0.0195
·FACTOR 12 ORIGINAL i.ciAiiJNGS
ORIGINAL scoRES 1274
o.o654 -0.0684 -0.0205
0.0383· -0.0219
-0.0386
0.7436
Oo041t2 -0.0334
x·-ii;f--0.0407
-O.Oit41
-0.0199
-0.004~
-0.0218
u
FACfOR- 13 OR IG INA.L lOAD-INGS -ORlGINAL. SCORE!i-1274- X--14·1·---------------------------------0.0it79 -0.0201
0.0121
o.0432 -0.0798
0.0268
o.0321t -0.080~
-Oo0413
Oo041t2
Oo7815
Oo0091t
-0.0633
0.0664
1~
FACTOR lit ORIGINAl lOADINGS
ORJGJNACstliiiesT271o-x--1~1------0.047l. 0_.0001
0.0003 -0.0683 -0.0590
-0.0397
-o.o196 -o.ons
o.oo95
o.nn
-0.028l
-0.0289
0.0174
0.0202
52
scores for the coefficient of congruence.
These results do
not indicate an identity matrix for this. case.
·•
Coefficient of inva:ria:nce.
Tables 16 and 17 present
the two solutions for measuring the relationship between
the appropriate set of factor ·scores.
identity matrix.
Neither table is an
In addition, the matrix of coefficients
\:of invariance based on Z, Table 16, is the transpose bf the
matrix of coefficients based on Z*, Table 17.
MoBified ·coeffic'ient of in variance..
Table 18 shows
the results of the modified coefficient of invariance when
!30% of the rows and columns of
i
1
.
z
are permuted.
Again, the
results presented in Table -.18 do not show an identity
I
.[matrix.
~~%
I
Row and Column Permutation
!n this case, an additional criterion was established
for an adequate measure of invariance; that is, it is·
J
expected that the degree of relationship between the two
i
1
sets of factors should be lower than it was in the
previous case of 30% row and column permutation.
Coefficient of' congruence.
resul.ts for· this· case.
identity matrix.
Table·l9 presents the
This measure does not produce an
A comparison of Table 19 with Table 15,
the results of the coefficient of congruence for 30% row
and column permutation, reveals that the coefficients are
lower for 60% row and column permutation.
Coefficient of invariance.
The.two solutions
53
·•
--------------·--------------------------------------·.
TABLE .16
~------------:---------~----------O.I!HliNM...S.CO.RES_AND. PERMUTED
------2
------··--····-
---- --------- ---·-···----------·-------
1 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X 1~1
o.9H6
o.0281
o.on8 -o.o259
o.o765 · ·· · ..:o~·on-' · ;·o.o16J
-0.10~~
0.0059
0.171~
-0.0663
fACTOR
2 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X 1~1
0.91)7
0.0092 -0.005) -0.0105
0.009)
o.0825
0.0~2~
-0.0390
0.0~51
0.0_888 . 0.102~ ·- ------~- -------~---. ···-··-·------··--.
~
j 5
·l
6
1
I
9
U
FACTOR
FACJOR
3 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X
0.0680 -0.0010
0.1835
o.o8o~
0.1391
0.0502
0.0639 -O.ll51 -0.0256
~ ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X
-O.OH3 -0.00,5
0.9320 -0.1316
0.15'9
-0.0629 -o.oo'o
0.0920
Oo128'
FACTOR
o.o~o~
~~·
7 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X·
0.0328 -0.0723
-0.0110
0.0588 -0.0033
O.l't~9
-0.0263 -o.on' -0.0716
_1~1
FACTOR 8 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X
. -0.0111 -o.o'o~
o.0598
o.o279
o.o138
0.0361
0.0715
0.1368 .~0.0577
1~1
FACTOR
1~1
I
'
I
1~
I,
-0.01~5
o~-0539-
0.0679
-----
- o.1oo1
.
0.0528
.
Oo0866
-0.1666
o.o~88
-o.o~ n
-0.0686
.
-0.156~
~
·l
-- j
l
-------------~-------------1
o.~112
0.1011
0.5931
0.0322
0.868~
0.0209
Oo5827
0.1123.
0.0066
o.zo81 ____
·•
I
-··------+
Ool261
0.5567
-0.2f>83
0.0612
0.6583
-0.1185
0.10~6
0.1709
-OJ2221
FACTOR 10 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X 1~1
o.o112
o.o129 ·-o.1o~1 · -o.o38a -o.i5~2- ------- :.o-.o872
-o.o,09
0.1268
0.062~
0.16~0
0.1~11'-
o.;o&53
-o.1o29--o-:.-7658-
FACTOR.11 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X_
-0.0536 -0.0095
0.0789 -0.076)
Oo052~
o.9056 -o.o2'5
o.o917
o.o293
Oo1561
Oo0561
-0.00~3
~0.0,52
-0.-0~08
0.0-985
Oo0855
Oo1961
FACTOR 13 ORIGINAL LOADINGS
ORIGINAL SCORES 127~_!1_1~L ,_
_ ___ _
o.to8l
o.1081 -o.1lot · o.o .. 99
o.oii50
-0.278~ -0.8398
Do0532 .-0.1620
Oo11H -0.1019 __
Oo1358
-0.0322
.-FACTOR 12 ORIGINAL LOADINGS
ORIGINAL SCORES C2n X
-0.0052
0.0515
Oo1~39 -0.0530
Oo067~
-0.0~29
n
~~· ..:o.o866
6 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X
0.0008
0.1251
-0.0821
0.0~09
o.ts~'
-0.0878 -0.1 .. ~6 -0.2267 -o.o2u
9 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X
.0.0811 -0.0088
0.0~70
0.0372 -0.00~3
·o.023l
0.1166 -0.0605
0.0015
I
DolO~~
1~1
FACTOR
-0.0~90.
-0.0015 .
1~1
5 ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X
0.0091 -0.0510. 0.1613 -0.1195
0.88H
0.0169
0.09~8. - 0.1 .. 91____ Oo2l12
FACTOR
l
----~
FACTOR
0.02~6
3
--------1
SCORES - 30% ROWS. AND COLUMNS. __________ ---·------- ·- - ·-· ..
0.90~9
-0.13~2
1~1
-0.0770.
_1~1_____
.
-0.1665
-0.075~
FACTOR 1' ORIGINAL LOADINGS
ORIGINAL SCORES 127~ X 1~1
-0.029'
0.0996 -o.ot~a
0.0623
0.1372
-0.0566 -0.0638
o.oz~z
-o.o516 ,-o.o989 ___o.uos ________________ ..
0~0992--
54
.,
COEFFICIEN'l'_OF __ INVARIAN~~:;___{Z~-------
1 ORIGINAL LOADINGS
PERI:CUTEO StORES 12H .. X_J~I_
o.91t16
o.oz.r.2
o.0681 -o.o242 · o.oo96
-D.D5~4
-O.OD51
0.1083 -0.0293
FACJOR
1
2
2 ORIGINAL LOADINGS
PERMUTED StORES 1274 X 141
o.0284
D.9137 -O.DD7D -D.D046 .:.o.o5D9
·
o.1251
o.D328
-D.DD98
D.0575
D.1D78 ... D.100Q. __________________ ~---- ...
-
j
FACTOR
6
FACTOR
I
,I
~-'- .
3 ORIGINAL '.LOADINGS
PERMUTED SCORES 1274 X 141
0.0717
O.D093
D.7397
0.1548
0.1674
D.t545
0.0788
Oo1439 .~0.1lD1 _-O.D147.
FACTOR 4 ORIGINAL LOADINGS
PERMUTED StORES 1274 X 141
-D.026D -O.D052 · o.1835
o.932D -o.t'i91 --·----.;;:D·.-oii2D
-0.0764 -D.D529
0.0498
0.0625
5
1
6 ORIGINAL LOADINGS
PERMUTED StORES 1274
-0.0183
0.0093
0.1044 -0.0866 -0.0144
-0.0771 -D.1665 -0.2784 ·-D.D566
•
9
FACTOR
11
12
u
~D.D723.
o.0586
-O.D4D5
-O•DD89
·o.D12a·
O.D599 .·..
0~0469
.--_0.1DU
o.D28D
Do0)69·"-.:.D.D389--
i
~~
X141
0.4372
D.D322
Do5827
D.1D45
-D.0872 .
FACTO~ O~D~=~GIN~~O~~:OI~~~ 021 :ER~~~~~4~Co(l~~(j~~~"--l'-..l~Li);\ol2- ~O.B684--::ci".-D4lB--O-~ 1706--c-0.1479~
FACTOR
.. 10
l
.. -------D.0818 .. D.Of7if·_··.
-
5 ORIGINAL LOADINGS
PERMUTED StORES 1274 X " I
.
0.0408 . -o.DD3)- D.Dl36 -D.DD39 -D.l543 .
O.D766 -0.0106
0.0803 -0.1319
D.8814
O.D52L . 0.061)_ ... 0.08..5L ... OoU_U· _ _ __
·---··--·-----------------·---
0.1562
i'i
-·o.D107 -.:.D.D1n
FACTOR
FACTOR
I
o.ooo1
-O.D4D8
-D.D398
8 ORIGINAL LOADINGS
-O.D64l
. ·- ---. ................ --··
-··-1
...................................
PERMUTED SCORES 1274 X 141
~=~~~= ~:~:~~ ~~:~;~: -~=~~2! -~-~:_D~~-------~-D-·~~~1----~·~2D7
9 ORIGINAL LOADINGS
PERMUTED StORES 1274 X 141
0.1211 -0.0013
0.0868
0.0492 -O.D69D
0.1[24
-D.OD4D
D.D857 -O.D322 -O.D176
D.2D83
D.556J
-0.2682
-~~~22~3 --~-~n~j
D.6583 .. ..:D.102a
FACJOR 1D ORIGINAL LOADINGS
PERMUTED StORES 12H X 141
··----- -----:---;;,...-;;;-;-;
D.0279 -o.D488 . -o.1665 -o.o47-7 -o."i563·----- ----o.oD66 --ci~-i258---o.D6lz -D.l186
o. 7658
-D-0452
0.1968
0.0994
0.1027
FACTOR. 11 ORIGINAL LOADINGS
PERMUTED StORES .1274.. X .. 14)
-0.1049 -o.D386
o.o5o2 -o.o627
D.017G
-o.o877
0.9056 -0.0429
0.0533
0.0244
!
----1
D.1448
o.o1a1
o.o228
-o.o408
.FACTOR 1l ORIGINAL LOADINGS
PERMUTED StORES 1274 X 141
----------------------..... __
o.oo58
o.o~t51
o.D639 -o.oo4D
o.o949 ----- .... · .:.o-~1446 -D.o26-.;;---- o.on6
o.U64
o.u6&
-0.0245
0.9049 -0.1620_ - . 0.05..15.................... . ................................................. ..
FACTOR 13 ORIGINAL LOADINGS
PERMUTED StORES 1274 X 141
o.nu
o .D89l -o. 1.352
D.o921. - --ci~ i496_____ =-o:-2267--.:o.o374
o.o916 -D. uu
o. 7148 __ -o.o989 .................... ...
D. 1367
-o.o604
;
J'
1
o.D62f-
I
FACTOR 14 ORIGINAL LOADINGS
PERMUTED SCORES 1274 X 141
-0.0663
0.1020 -O.D256
0.1282
0.2172
-D.OZ41 -Do0714 -D.D577
D.oo1a
0.1641 ·
0.0291 -O.D755 ....-0.1018~ __ D.8506 ________________ .. __ _
........ --~-- ............... ·--- ----- ..... , ----J
55
~---·-·····-··-··•······-·····--------~-------------------- ____________ ..TABLE 18 ___ . ________ . ------- ___ -------------------------------.---------------
'
~----------··················-·-·----ORIGINAL.SCORES.AND _PERMUTED
'
1
FACTOR
1 ORIGINAL L(AIJJNGS
SCORES.
~. 30~ . ROWS AND __ CQLU~!l_ -·-<·--·-··-------------------
__ _
PERMUTEC SCORES 1274 X 141
i------------------··.;:&;f-~6~----=%;i~~g---=-t~lJ-:---~g~-a~-J~--:··-~---~~~~----------:-~~_t_!PL.:.'?!.~~~~---=--~!-~~-~~-----'?~-~~-~~-----~-~~~~-1-/
----- i- -- T.KfrilC. ·z--n Jff(;-nfit-·U:AoHiGs··--- Pf IHi\ft tO- S<!tA Es··c 2110- )l··rn··---·--------------------·---------------------------------------------::-ic>.-!.~2ri~'-;6o;3c_-,<Oe!..?;II2SI
' -.-. J -- . --F
-0.14117
-0.11)15
-0.1409
-0.0493
-0.0416
-0.0655
-Ool3'i5
Go0200 ,
A-Ctiiii-~~-~~~:-;GTN~-~~~;iffN_;;~lR:::·~~:c olll's-f2lCinH····----·---·-----···········---··---- .. -- -----···------- ------~
-----·· ·-·----- .... ::g.:-~H~---:k;}6;-(-··: t%~}i·-·;.-6: -~ ~!~- ----~!-~~'!?___ ---------.<'.~P-~!>-~___:_~!-~~-~~-----~!.!~-~~----":~-·-!~-~~----:~-~~-~~-~-1
4
FACTOR
4 n~IGI~AL
LCAnJNGS
PfR~LTEO
SCORES 1274 X 141
-------------------=Q,H.U ...::.0,_l!'.U.... .!!~_u}t ____ ~-~~-B9.'t. ...:!l.•.?.l19.l ..........:9.!"l!Ll.L.:-.'1!.'11l.ll_~-----'1!_~?.!!~. ----~-·.!Q.~~---.:~.!!.~?L.
-0.0079
-0.0116
-0.0841
0.0146
.
.
~
.
---- s·---,,Ac-riiR--~~~~-:!~;-,~-~~-;~-~~-oilif~~~-~~-"R~~-~r~~-!t;"oii~-~~-~~-r~--x-·iH=~:~-~~~---=-~:~~-;~-----~:-~~-;~---=~~-~~~~--=-~-:~-~:-;·
-0.~223
o.C373
0.1e68
c.0684
·
----c,- -- -- ;:·Ar.YoR-~- 6--nR iGit·fAr~i-CADiNGs- ---~F MMU T"F c··-sc OR if S*- c·zr~--x--i~l------~-----·--------- ----------............. ~---~---------_---·------- ··, ·
0.09~9
-0.0493
-------- -o.11ry1
Yoio ··-.:·o:-23-73--·::
o
·.:o~
~-i
........
~O.C348
ifo3-- --.:a. at 11;- --- ·· ------------------------------------- ----·----- --- ----- ------------------· --------·
-0.0214
-0.0373
0.3363
-0.0217
0.5117
0.0396
.. !-
7
FACTOR
7 ORIGI'IAL LCAOJNGS
PERMuTED SCORES 1274 X 141 _
.
._. __ .. ___ . ________ ~-: 9.• Ln ~_I! __ _.:_12.!'.<. _1_L .::.a • AE 5___ ::-_C._QC_H 6___ ::Q-~ () _8_1 J.. _.. ______:Q~~-~!.L ...'1!.l~ ~~ ... ::'.l1!.l1~_7_'!_____~_o_!!.!~----~-~~-7_~_8___1
0.1111
-o.o410
-c.028C
-0.1142
·
:
--'. ii---- Flie Yiiii---!i --ri itf(;-fiilli:. -i:c.iiiH·iGs··- --- i>f"li li ui EiJ - sc eli E·s- T27/;- ·x··.-~-i --------------------------------------·--- ----------~--------- ---·
•
-0.1073
-0.0654
0.1001
120
0.0252
0.0031
0.5117
-0.0.481
0.4498
-C.2036
O.C161t
:
7
~----9·-·· -Filci-nR--~-~~~~-;G.i~;~~;:~~iiN~: :ER::; :: :ccliEs -.-2-74--li-Tio"i ____ - -- ------------------------ ·------- ---------------------;
I. .______ ----_;~:~!~: ..-:~:H'i~ .. :-g:~~~i _-~: ~~:; _-0.120 1__ _---- ____ 0.~39_7 .. ___Ool_216 ___ ::'0o_203~- ____C!50_J_5_ ... .":~~~-~6.'! ..
l
10
FACTOR 10
ORIGI~AL
LCAOJNGS
PER~UTEC
SCCRES 1274 X 141
1
~--i:i: ....F4croR--~~~~:~~~~;~~-;~~o:i:!~~~~-!f"R:-~;-~~~-!co:.:~-:::~;-·x·Tio-i~~~~~-~~~----~:~-~~-~---~~-~~~~--~~~~~-~~~~~=~~~~=~~~
'--·iz·.---F-ACTOR~!;~!;;G[~:;.~:-!~olN:-~~-~~!~R-::~:.:!!cc:::~-::.:4..~-l~r~-~~-~~-~~--::~~~----~-~=~~-~o.o~~--~~-~
145
:
C.0625
C.2C71 -0.0117
0.0374
-0.2373 -O.Cto1C -O.C631
Col327
Co1t61 :
·-·--·· -·---; o0.0357
a·32· ----c-.-s
634"- ·-;;:(,-;I 64 a···; c;r:-s z·s ---------------------------------------------·---------------· --------------------------:
~o
.
.
I
FACTOR 13 O~IGINAL LCAOJNGS
PER~UlfO SCORES 1274 X 141
·
0.0,25. 0.'1233 -C.C704 -(1.0840. 'Q·.te67
-0.1803 -0.0280
0.1206
Oo0252
O.CC23
· ·· -· · --------- · ··n·;rflfl ",----;;: o-.-t6it a·-- --o ;5 'lSI.___ ·;;;o·; t:i9 !f.,---------------·---------------------------------·--------------------·------------------·
13
.
.
. --~it .. - Tl\ctnif T4--diffci"J.rilli"C ·n-Ail I~ Gs··-- -PERI'I.ir!O:o· ·st"t:lRt s--12 71,·· f<Tn··-·------ ---------· ----------- ------·--- -- --- ------------------•
-0.0823
-'1.0'130
-C.C766
O.CI45
. 0.0683
.
-0.07H
-O.l14C
-0.0479
-O.OitC9
Ool021
•
56
I
,·I
__
1
FACTOR
l ORIGINIIL LOADINGS
ORIGINAL SCORES 1271t X lltl
o. 33'H -o. o5·o·..". o. oz~to -o. 049o. · ;.o~·o733. -----------,,-~!i469·---o-;(,-3ii9---.:o;o:s.z·z·--.::o;o7·i;;--·--o;·o675-- ___ · - - - - ~(1,_()~88 ___O_.I.Q1 5 -~ O~_D67~--=0.!..I-.,Ooc4,_,3,___ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __:__ _ _ _--!
L
2
FACTOR
Z ORIGINAL LOADINGS
OPIGINAl SCCRES 1271t X 11tl
3
FACTOR
3 OPIGINAL LOADINGS
OR.IGINAL SCORES 1271t )( 11tl
-~: g;~ ~ _g: ~~:: ::g: g~ ~ ~ -· .~-g: -~~;t ~~:~~~~~~~~~~~~~~~=0-~~:~~~~~~~~~~~~:~~~~~:~~~~~~~~~~~:~::~~~~~~~:~~~~~~~
i
~-~.- ~~::R~g:g;::G~::~ g:::D~::~:~:R 1:::~:~:~-~::-:~;~;:-~--~-~~=0 .0~~~---~~-~~-~~~----~~~~-~----~:-~~~-~---~-~~-~~:~--1
o"
o.,
;.o; ·en
-D. 9o -o. o 579
o. OA45 .
o6 1 · · ·
.to ----------::tr;oolr---<>;lio6"J--.:o;oo&r-·:::-o;·olfi6---::.·o~TPis__ _
______c_"Q.1!l__ .Q.._o_3~1~ _ __o. 01_85 -o.o=82,_9'----'---------·- - ' - - - - - - - - - - - - - - - - - - !
5
FACTOR=
--~ ___ FA!;_T.QR
7
g~g~i~ ~g~ g~!t ~Eg: ~ rI;~ ~g ht~:~~:-~~:~~:~:~~~~~~~~~~~~~~~~~~~~~~:~~76:~~~~~~:~:~~~!~~~~:~:~~~~~:~~~:~::~~~~:j
G.
6 ORIGINAL L.OADINGS __ {)!\JJ:J.NI\.L S~..\C,JO,!!R~E~St,;.{l~2f7!!1t_.!!.X....&:11t:u.I-;;--;:-;~;---;;:-;;;=;----;~=-;;--;;-:==--;;-==C.O't69
a.0627 -0.0046 -0.0036
0.0175
O.lt418
0.0001 -0.0159 -0.0349
0.0583
FACTOR-O:
1
O:::GI:::
O:::DI N:: OOO:R
fG-~:: ~~:~ ~~~-.S--~-;~:-~--~-~~------------------------------------------------.-----------------1
o. ol88
o. o3 As - o. o2 ao
o. oo64 ·
0._()_16_4_ __ 9_,_.Ql2_5__ ~0_._(10~4____-_Q.OOll
I
:.:o .·o7az·-----------o~oocrc·--o-~nn---·-o;cio'f6---.::o;ccr7---.:.-o~-o59i---1
.
r~;~TOR-0.0521
R OR I Gl NAL LOAD! NGS
ORI Gl NAL SCQII..~~--1-2_I~_l(__ !!!L_ ________________ ._ _____________ ,__
O.OORR
0.0621t -0.006·1·· -0.07't7
-0.0159
0.0075
O.lt876
c ______ ... ---------------------
o. D026
·
10
-o. 022 8
0.0305
0.0960
- o. 061t6 _ o. oz6o __ -------------------------------·---------------------------------·----------------------
FACTOR 9 ORIGINAL (OADINGS
ORIGINAL SCORES (274 X 11tl
· · · - o. n 1 'I -o. aHa
o. o5 47 -o. 164-- -o--;Ob-35 "-='-=-"'o'.'o"'3""'"o.-=-"o-."o"H'"'7y--,o'.'o"3"oe<6--o".""to"9"5"9.-:.c-..oc-."o'"z"o,_........
o. OR 58 -(). 0356 -0.181t2
0.0 132 ...... --------- .... ------------------------------------------------ .. -----------------------·
o
FACTOR 10 ORIGINAL LOADINGS
ORIGINAL SCORES 1271t X 11tl
o.o676 -o. o413 -n. 0193 -o.119, · .:.o~ozz9·-----------o-:o5-at.----:=o-~c)59z ____ o~o959
o:o;,z-o--·--o~-3~6~--
---·-·---ll_...~!C.Q.._::o.osLL...._Q,.!!_lo'l . - o . p " ' 5 " ' 1 ' - ' 2 ' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1
ll
I
.
FACTOR ll ORIGINAL LOADI"'GS
ORI Gl NAL SCORES I <:.0::~__!1(__}_~!__- ________________ ----------- ___ _
.-('1.05'10
o.0397 -0.0644
o.on2 -o.0703
-o.oo89
o.OI63
o.oo21
o.oe56 -0.0100
o. 3 6 B4 . -_o .1_ 0_2 (]_____Q~_QH.L. _-:Q.&4.I?. ______________ ~_:_ _________________________________________ --~-----------~
FACTOR 12 ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X lit'
o.1o1s -o.r-CI12
o.oo56
o.o351
o.oZit8
-o.oa61
---------- ----·-- ::!:'.•J_Q?.CL-... .9.~ 3f>~J1. --- q. 0_5 J.'! _____<h.O.!l!>.~-----~---------------12
o.o12s
-o.o229
-o.0356
-o.o5n
I
·--------------- .. -------~
57
presented in
Table~
identity matrix~
20 and 21 also do not yield an
A compariso~
of Tables 20 and 21 with
·•
the results of the 30% row and column permutation case,
i
I h
• Tables 16 and 17, indicates that permuting 60% of
is
own 1.n
I
!the rows and columns .of Z produces lower relationships
i
~etween
\
.
the factors than permuting 30% of the rows and
, !columns
of Z.
\
Again, one of the solutions for the
)
'
. ~oefficient
of invariance is the transpose of the other;
(cf. Tables 20 and 21).
Modified c·oefficient of invarLance.
Similar problems
were encountered in this case as were·true for 60% and 90%
:column permutation.
That is, the empirical results were
!
:unavailable.
Further implications of the modified
i
'!
•
.
!coefficient of invariance.for the row ·a:nd column
permutation case are presented in the summary of results
)for
th~s
section.
.
I
,--
1100% Row and Column Permutation
----
In this case, all the rows and columns of Z are
!permuted to form Z*.
The results predicted for an
·,
adeq~ate
!measure of invariance were that the correlations between
the appropriate factor scores for each coefficient should
ibe due to chance alone.
I
In other words, the relationships
I
!between the two sets of factor loadings have reduced to
I
i
!that expect.ed by chance; therefore~ an adequate measure of
I
!
.
:invariance is expected to reveal lower coefficients in this:
'case than those presented for 30% and 60% row and column
,·.
,
58
.,
-- .TABLR 20.---------------------------------------------------------------- -----~
'------------------ ---------------------------. ---- ___
~--------------------"------------------
-
ORIGINAL SCORE-S AND PERMUTED SCORES --601. ROWS AND COLUMNS.--------------------·------------,\
- - - - - - - , - - COEFFICIENT..OF-.lNVARlANCE..-.
-~"~~-~==~-~-----------~
. -1--FAC-TOR-Ci.Jt>J'Gfi~-~C LOA·o-i>iG_S_ oiffGiNAL SCOR"'E"'S"C"2"7"1t"'x,..-,1-.:lt-.l-----------~----------!
( ·
o.o~o 78 -o. 1 0'>5
o. 2771t
0.2808 _ : !?.!_o_~5_! ___________-:~-~~-2_~!>_ ___-:!l.~!_'!~-~----~~.?._'!~_o_ ____ !?_~_1_~!!_ ____Q!_l_~-~0.~653
-0.061t7
0.11t20 -0.0983
2
FACTOR
-:----
l
i
3
2 'lll I GIN~( lOADINGSOR I Gl NACSCORES'"l"2'f""liT4l-----------------------------------------------------------:0.03qo
C.HCO
0.2734
0.2595 -o.o~1'1
0.2765
0.0612
0.2182
0.1009
0.1531
I
--------'0~2634.
t:ACTOR
---n; 152l----;;o;n·n-o~ nz
3 OR I r.t NAl liJADI NGS .
_
_g:~~:: -~: ~zr~ _g:~;:~
1
·sc'ciRES-Ti7~-~Cr~.-----------------------------.--------------------------i
_g:~g; !l.~.l.~!.~-----------~-~!lc1_~~----!l-~!-~!.!...._-:_o_!_o~~-~-~---o_!_z_!Q~---=--Q!_Q~~--j
-OR I Gl NAl.
....
-4-F-AC-TOR""""4-riP flrt'NA'i:.-(hAOiN·G·S--ORIGI"'Al SCORES 1271t X 11tl
0.1522
o.0090 . 0.0974
0.0207
o.OZ17
0.3526 -0.0786
0.1031t -0.2300
-0.1700
o. 1373- 0~ 1111: 0.3702
0.1 78'7"·-------------------------------------------~------------------------------- -------,
>
-5--~-~C~~ ~~-O~O~: ~ ~ ~-~ci~ 1~~-!01 N~~_3 7_1t_~R i ~~ ~~ ~~~COR~ ~-~!-:~-,.--x--r~.--~-:~~~~----~-:~-~~;---~:~~~-~---~~:~~:~---~-~:-;~~~--i
0.1208
6
FACTOR
0.1121
0.2232
:-,o--F'AcroR 1 o I'JR t'GTI'i.\l L oi.o1 NGS
·r·
. T1
I·
1
0.0407
6 OR I G I NU U:iAOI NG S
·ofif GINA i. -SCtlRES--(-fji~-~t"T~f--------------------------------------------·----------------0.0624
0.2913
0•1748
0.0856
Oo1367
0.2785 -0.1611t
0.6036
0.1258
0.0567
-o
.0111 -o. 1 77.2 - o. 1211o
o. 0606______ --------------------------------------------------------------- -------------------
_g:~:~~
oR 1 G1 N'"'A-=-L~s-=-c-=-o-=-Re:cs;::--c'"'2""'7'"'1t,.....,x:--::1-:lt-::l----------------------l
g: ~ :,:~ · ~: ~~:~ -g: ~!:;
-:-O.(J_,..I!_Q __________ .::Q .._QII_9_l_____ (J_._o7?~ ___-:-Q,(J_QO_~ ___-:-Q._l(J?~ _____ o._~?9,.__ _
-fA fro·..-. 11 '~", r.-ii!A ._--- L.oiollic_s ____ £1R:i'GiN'A'l_s_coi!e-s--127~-XT,.-,--------------------------------------------0.0139
Oo20S4 -0.055& -0.07\18
0.01102
-0.1079
Ool201t -0.0885 -O.ll21t -0.12S1
0.6922
O.OlS2 -0.0032
0.0457
L-----------------------12
fACTOR 12
_c __________ - ------ ------------------------------------
OPit;IN~l
0.00'>6
o.0279
lOADINGS
ORIGI"'Al SCORES 127,. X 1,.1
-0.1127
0.1H2 ..0.0444
0.0812
-0.1600 -0.1093
0.0'197
0.1146
0.320~
o. 8065 -o. 1787 -- -o. o 012------ --------------------------------------------- ·-- -- · ---- ---- -- · ------------·
59
;_ ------------------------------------------------------------------- -- --- - TABLE . 21.-----------------------------··----------------------- -- --------~
;------------·------------------------ -· ORIGINAL: SCOREs.AND PERMUTED. SCORES.. ..,-. 60l ROWS- AND_ COLUMIIS __________ __:_ _______________
------- COEFFICIENT- DF-~NVAIUANCE-. .{V
-I
--.--FacfoR--t-n!!TiiliiiiL Lniof~iis··---,.eil-,..-uTEn scoRES 1211t x
o. 'lt78
().0141
o. 0378
0.0057-
0.1398
0.1870
J
--------~
1~1
0.1524 __ ::-_o!_03_67___________-:Q_~I?!!.?!!_ ___-:9_.!_>_~~-"! ____ _c)_.U9.L. __I!• O!~!_____O!Jl~~L_.
-0.0977
Z OR IGI NU- LOAf)J'iGS
PERMUTED--SC(iRES___fz7~-x--l-~"f-----------------------------------------------------------0.1054
0.3700
0.1976
0.0091 -0.1568
0.2913 .0.0204
0.0331
0.0266
0.1888
.. FAt:TflR
~
L'"'"'-~~:ll\'' ·:~~\li"':E::::···:~!~i' "~~.:::~:·~·~r:;~~~~~::-_;;;:;;:;:::~~~;~;:: .:~;,;;:: :;~;;~q
----------o~?os'i-·..;o.Hze
o~T5211
F-ACTOR~OR-tGTNAL LOAOINGS--PEil"UTEO SCORES 1271o X HI
4
. o. 2so6
-o.o798
o. 2594
o.oto4to
o. 2111o
o.oo85
o.1102 __ ::O!_<U_'19____________9_·!l.!I.H ____ g_~~-loz.~ ___-:ll_.1_~H-~ ____ .<>..1~9.~ ___ :-_Q.n_I!.L_ ·
o.1292
·
FACTOR 5 OR I G I NU -LOAD! NG S. - PE RI'UTE o -SCORES___f2'fi;-x--i~r------------------------------------------------------------.
0.0557 -O.OA20
O.l571o
0.0206
0.5993
0.1166
0.3475
0.1458
------o.oaoz
(). o8iT-6. 1"698 ·.;o.oi72
5
.. FACTOR
~
flR I G I 'IIAL LOAD I 'IIG S
-0.~242
-o. 1081
1
-:.-o.tnr-
Q.2765
-o. 1 60C
PERMUTE o·-s(ORES--(-;Z'fi;-)fT~i-----------------------------------------------------------------·
o.n115 . 0.0216
0.1331
0.2785 -0.1~60
0.5618
0.1210 -C.C891
-0.1028 ·- : o; 0419. · ---· ------------------------------------------------- ··---------------------------------·
_
·-nt--roR~-~?;~:~-, N6~~~?!o' N-g~ 1-;-~~E!I11~ ~; ~ 2 ~c-~~-~~~g-~~ ~--~-~~:::.()_._!~!~ ____ 1)_._?~?~- _:::.P_. ?-_1! ~-~----() •?-_0~-~-----I!._I!HL 1
0.12u4
-FAt TOR
9 -
FACTOR
-o.o339
o.1958
I
8 OR I G I NA·C-LifAiil NGS
PER MUTED SC-ORE~C!":z7~YI~i---------------------------------------------------------0.2421
0.2180 -0.0445 -0.078&
0.1401o
0.6036 -0.0655
0.2961o -0.2408 -0.0006
--------·-.;;o~·os87·--o.o996
o.1608 · o~i-;z-o·i. ·
1
8
-o.1o9z
10
<; 0~ I GI NU LOADINGS
PERMUTED. SC-OR{S-T27~-X--i~i--------------------------- -------------------------------------n.l'JS6
o.1010
o.2109
0.1015 -0.2264
o~_1?.~~-----o_.z_zz_11_ __ :::.()_.~_5to5
o.49~-~---::-_o._l_~?-_L_
-0.1122
0.1148
0.0312 -0.0780 -------------------------
FACT~R IIJ nRJGINAL LOADINGS
o.139~
o. 1531
PE~~UTEO SCORES 1271o X llol
-o. 0646
-o.z1oo _ -o.• 1•n_~ ____________o_.05_6_8_____ Q_._Q89~ ___-:o.9_1o_o ___-::9•_180~-----~·-!!.9_~~-- .
f-,' - ',,,,.-:;·:::'.•::·:::•.•:;'''~..~;;:·~...-.--.-;;;-,---,.,----------------------------:-~---1
'-u---~FA-tio~--:-~~~::crN;~~~::oY~!~g-~~-:ER:;~::~;c_o~::-~:-::-~xi~T~o.o772
-~.ot:47
0.0153
-o.1522
0.8065
-o. 0461
-0.2025
. 0.1001
-0.1258
-o.oo51
-0.1852
,
_ 6.1373_____Q_._u_~~---------:I)_~!-~~L-;Il.!.!l.~!~_: __ _c)_,g!>_~----O.•J._Q..~!! _ _
G4i!.H__
-0.0163
.
-~-3--fACTOR
T3-riR jjffNAC-LrJ-AOJN-GS
PERMUTED SCORES I 27io X 1lol
o. HI B -0.1392 -o. ZH 1 ____ 0.1lll _____Q_._~?_H___________-:!l_~!D~----=9-•9-~~-~----Q_'_l_~_l_O ___ -::_D,_(l~_3_8_____o_,_o_!~L-.
-0.0012 -0.1787
0.421o6 -0.1277
l
lit
FACTOR
-~~O~= ~G1 N~~ l ~~:01~g~ O~;;e RM~~~r~!COR~~~r~r~---~~-.--~-:~:~-:---~-:~-~~~----~:~~~~---~;:~~~~---·-~:-;~~~-~~
------·o.o~ss--~o~oifn·----o.o9o2-o~-i258
1
60
permutation.
Coefficient of congruence.
The. results for this
measure are presented in Table 22.
Table 22 indicates
l
l
[practically no relationship between the two sets of factors
!
The range of
coeffic~ents
of congruence extend from -.0004
to .1329.
.lI
Coefficient of invariance.
Tables 23 and 24 show the
I
two solutic;ms for permuting 100% of the rows and columns of
Z to form Z*.
These results indicate that the coefficient
of invariance does not reflect consistent relationships
1
.
between the two sets of factors.
In addition, Tables 23
i
'
iand 24 do show relationships which are lower than those
i
. :presented for 30% and 60% row· and cblumn permutation.
·i (Cf.
Tables 16 and 17, ·and Tables 20 and 21).
Again, one
!
bolution is the transpose of the other.
lcoeffi~ients presente~
The range of
in Tables 23 and 24 extends from
I
1- • 0 0 6 8
I
to . 5 916 .
Modified coefficient of invariance.
Table 25 presents
l
the results for this measure.
These findings indicate
[that the relationship between the factors is lower in this
I
!Case than it was fo~ 30% row and column permutation~ The
I
iodified coefficients of invariance range from -.0027 to
1.4569.
~.I
!
Summary of Results for All Five Criteria
None of the empirical findings for the three
measures of invariance produced an' identity matrix for any
61
-------------------------------------------- -----,--SCORE_S.AND
·- _______ -------------------- ______ c • .01\I GINAL
PERMUT::~O:~-:=-~~~-:~:-~~~~-~~-~~=~::::::::::::::~::::::_:~:::::_ :~i
COEFfiCIENT OF CONGRUENCE
FACTn~
1 ORIGi~AL LOADINGS
ORIGINAL SCORES 127~ X 1~1
-o.o764
~- oos~t
o. 0456 -o.o4r,s -o.o112
1-------~-~~=o"-~
o.o1e1 -,0.1298
o._«?_1J~---------F~CTOR
__ 3 __
FACTOR
--
~
3 ORIGINAl lOAOINGS
X
1~1
ORIGINAl SCORES 127~ X
4 ORir.INAL LOAOINGS
-o.o445 -o.o1o5 · o. oRn -o.oo1o -o.1oa3
0.0~42o. 0148
0.10'>9 -0.0181
1~1
ORIGINAL SCORES
_i;:g~~~ -~g: ~::~ --~_g:-~~g
~-
FACTOR
'
127~
'
-~:g~~g :~:gb~~ _:g:g!~~ :g:~~!~
6
j-__- _---_·
fACTOR'
f·
---o~o3'to···;;:o;·o5oo--·--o.-o•roif-
g:~!:~ •-o~0256 . -~ -~-~~::6_~~-~-----~·o_ao_~-----~~-~~-~~-----~:_~~~----~~-~~~~~---J
-o •.o5o9
r---,~~~~~- 5--~R~~~~,.-:-t~~OINGS-- O:IGI~AL SC~~~E~ ~-~~-,-~--;:~----
·I·
- o.o1n
2 O~lr.INAL.. LOAOINGS
ORIGINAL SCORES 12n X l~_l
0.0054
0.0138 -0.0781
-0.0306 -O.~lt17
-0~1030
0o 024lt.
0o ')079 -0.'0463
0o11J7
.. _ ... _____ . ______ ---•--··--------·-------------------·------C------------~
2
1
o.oo~2
--o.o759
-o.1o19
I
-o.o21,.---- o~o'fot.···::-o-:oo93·-·-o.:ii1f.-·-"
-i
i
---~~o~l>~ -~-~~:~s~~~::::.~~~~~~:::~~::o~:~~::_
1
6 0~1-GINAL UJAOINGS
ORIGINAL SCORES 12H X 1~1
··· o.-1o3c · -o.o562 -o.o5'9 -o.1o19-- - ·-.:.o,;-o,-:fi--o:-o-o2a ,-0.04U
o.oon
o.0691
Oo03-11 -0.0438
0.0014.
0.0961t
-------~ ____________________ : _______________ .:. _________ ,
o.o~42
1
FACT~~ 7 ORIGIN~l lOAOI~GS
O~IGINAL SCCRES 1274 X lltJ
O.OI1'l
0.0421
0.0800 -0.0276 -0.0'164
0.0028
__________cy~_o_2~ ___--:'l~_'l~l_?_ ___ .:~·_(J6_3'1 -o~o_2_57
0.0305---:;;0~0n5·---0~-655~r-·::o~-~07---·
I
r1---.
9
1
¥0
[-~~-~- 12
fft~'"~_ g~ g~Hr, 1~g~ g~~~o-' :g~ g~:r, GEg~gr ~~~-o~~!~ _x 1•»_ 0_. 0~ 4 ": _:~o:· ~:~~~:::~:~:~~~~:::::~::~o~~:::::~~:~~:~zJ
1
9
FACTO_P -O~O~~~ Gl~~~ ~g:OIN~~ 09S~RI~~~~~ ~C0Rg!o~~~4 _ll__ Hl_o;oim
-0.0213
0.0125 -0.0046 -0.0092
0.0555
0.0023
-0.0065
-0.0163
FACTOR 10 ORIGINAl. LOADINGS
QqJGINAL SCORES 1274 )( Ui
o. o5o~ -o. oan -o. 0667 o.o 114 -o.o353
o.o69l ·-·..:o.o'>o7·-·-o;ossr·-..:o;oi&3---.:.o~-f~>Z6
_ _ _ _ _ (l_o_()_l}'!___0~046s'~Q':'_I!__-D_._QO(J'> _________~-•----------------------I
121~
FJICTOR 11 OP!GINAL lOADINGS
ORIGINAl SCORES
X l4t
I
-~.1329
1).0245 -0.0310
0.0342
0.0030
0.0331
o. ozio··-..:o~05 io··-~o:ozn·-·-·o:·on;.·--·
-o.o1oo -0.1263
n.l151
-0.0487
FAC;~R --1,: OP.-1~-; NAL -lOADINGS
0~-; Gl N~l SC-~~ES • I ~7~-~--14; --------- ----·--- --------- -- ·------------------- ----0.0781
o.on79 -0.0118
o.014R -o.0063
-O.llf>l___:?~_()_4_()_3_____ 9~007t._ __ O. 0266 ___________ _
-0.0438 -0.0012 -o.oo49
- . - ------ _____________________________
FACTOR ll ORIGINhL LOADINGS
OqJGINAl SCCRES 127~ X_l~i
-Q.lH8 -0.11462 -0.-071o8
0.10'>9 -0.0630
0~0074
- - _D__._l__l~~ ___0._01)16 -0.()!!9__ 0 .• 076_2
----------FACTOP
1~ ORIGINAL LOADiNGS
ORIGINAL SCORES 1274 X
0.0190' '1.1137
0.0490 -0.0180 -0.0640
-0.0'>87
0.0266
6.0761
0.0584
1~1
-o.o639
.. __________"_________ _
o.Ol26
o.0'>68
-0.0659·--.::o.oo~l'---:.-o.oou __ J
-o·;ozoc-.:.o;oo9r·.:.·o~oo~--~
.l
62
·•
. TABLE 23 .•.... ---· --------------- --··------------------------------ ------------·
. ORIGINAL SCORES AND PERMUTED SCORES _,.__ 100\ RO'HS .liND. COLUMNS. .. --------------------------------~
_COEFFICIEN~~ARIAN~~~----------------------------------~
---c--a--··FAnoR-- -l ·oo fG.I NAL-·l-o7il5fNGs- ·-OR I Gl NAL scoqes I 274 -X-Y410 •• 151
0.0765
0.1717
0.4741
0.0139
0.3188
-~.0840
0.1689 -0.0875 -0.0163.
2
2
fACTOR
ORIGIN~L LOA~INGS
ORIGINAL SCORES 1274
0.~765
~-1293
0.1~54
-0.1614
0.013~
0.0378
5.~ Q._QR~_Q o._~'J_8'L
_____
_____
_,
X 141
0.1014
~.1107
o. 35
-0.1700
~
0.3333
0.1310
o.29o6
. --·---- .. -·
0.3104
0.1763
0.3153
I
- · 4- - FACTOP.. - -,.- riRJGJNA_L_i.n:.\ii!'lr.s-·-ORIGINAL SCORES- -, 274 X 141
o. 3740
0.1328
FACTOR
-3.
~nt.e
0.3583
-o.·o299
0.1361
-0.0643
-0.1621
5 ORIGI"lU L'lAOINGS
~.~420
ORIGINAL SCORES 1274 X 1'tt.
.
0.4326
-0.1564
o.45R8
-0.040~_
-n.zz9o
o·.znf ·- t.286~
·
c.na,
6 OPIGINAL LOADINGS
7
-0.101,0
0.1871
o.orz7
n~IGINAL
SCORES 1214 K 141
r.nMl
o.to311
o.11'14 -o.o2~11
o.!W34
-o.3oll5
1
-f"Atf()q O: ::: Gl
l:::OJN:: Oll:R-1 G::::B:CORES I 274 X -fi;j___
N:~
:g: ~~~~ g: ~~~! _g: i!:~
FACTO~
g:~~::
o. 38t,1
o.0082
::::: .~::::~: ~:::~::··~::::::]
o.Od~>O
·
11_ •.4c:Jlll_
0.2520
·--------·-------· ...
_.-~o.o_s~-6----~~~~~~~~~~~~~:~~~~~~~~~:o~-Q2:67::1
OP.IGINAl lOADINGS
ORIGINAL SCOR[S 1274 X 141
o.2548
o.o6Q4
o.n652
-D.1290
o.2573
C.3626 -0.09o5
--------------
-0.1617
-0.2028
r-9---=~~0R- ~:O:::GI N:~ f:~:OIN::Jll:~l ::::: :CO~~~
-0.0822
-0.1664
. 11
12
ry.0200
0.1~03
0.1732
-0.1198
0.4003
0.0213
0.1141
12;4 X· 141
-0.2522
F-ACTOR-10 ORiiffN~L LO~iJJNGS--01\YGINAL SCORES 1274 K H~~-------------------------j
0.1750
O.li81
. 0.1198
-0.1989 -0.0657
-0.0941 -0.0765
-~.27Q7
0.1730 -0.0~4~
0.0887
----·'
-------------------~----- FACT'lR .11 ORIGII-;ULOAOINGS
ORIGI'IAL SCORES (274 X 141.
-C.I195
c.2r.12 -o.ot74
o.1268
o.1851
o.o808 -0.1795 -0.279..
0.15 .. 5
0.0373
0.1954 -0.0490
0.2749
0.1339
--------- ------------ __________________________________________ ,.
FACTO~ 12 ORIGINAl T!Jt;llrNGS
01\IGINAL ·sCORES (27io X 141
0.117~
-~.1756
0.3737
0.1890
0.2243
0.0951
0.0114 ____0.1575
0.0635 -0.1034
-0.0087
0.2700 -0.1012 -0.1315
··-o--·-FACTOR 13 OQIGI'I4l lOADINGS
O~IGI'lAL. SCOR(S·-·(27io.. JC"f41
D.0339
n.1046 -0.1212
0•0901
0.1116
o.ot .. 7
0.2645 -0.1891 -0.14~8 -0.1018
14
---~
~
9
f·-ro·-
o.ton
FACTO~
1 .. ORIGINAL LOAOINGS
-0~096 . . . D.2289
0.1463 -0.02.54
SCORES (27~ X 141
O.llf.6 -0.0838
0.0506
0.5916
O~IGI'lAL
-0.0232
.-0.1107
-0.1614
0.2669.
::0-~014
__O.(J8~_f! __ J
I
63
.,
___________________ _!~~.1!-~-~----------------------------------
·------------------·
----------------- OR_!~-~~~~£Q~~--~~--~!'.13!'1.f!!'_I':~--~£Q~~--=---!-.l!~.!_R._Q!'!!!__~-~-~~~_!l--------------------!
--------~-------------------------COEFFICIENT
·-~--
2
OF
INV~!~~__i!!l__ __________________
FACTnil -t-nRTGfN.\CtfliHiTNGS-PE q MIJTEO- SCORES -(2H X
O.'H5l
0.1014
0.0793
0.3742 · O.Oit22
-('.1395,
o.u 75
o. 0337 -0.0963'
f•H
0.2161
-0.1039 _____ q~.?~.H.•• =9.•_Q_8_!J_ ____ ()_~!J:~~----l
7 OIIIGIN~l LOAOII'lGS
PEq'4UTED SCORES·I274 X l4t
~.0765
0.1413 ~0.0066 -0.2296
0.10_3_!__!)_._2_6]_8
~0.17'56
0.1046- 0.2290
FACTI''!
0.~764
0.2072
'
C~ 1 ~~~GIN~~ 1 ~~~01~~~244 ~ER~~~~~O~COR~~ 4 g!4 X i4t 0 ~~ 3 ~:::::~;~~~~::::~~~~~:~::::~:~;~~::~::~::;;~~:::-~
~----------=-~--!'17_3_ _c.3~~7- -0.1~2 -0.0212_
________
I
3
FACTO!!
4
FACTOR
4 ORIGINAL LOADINGS
PERMUTED SCORES 1274
1).4739
0.1852
0.0808
-0.1040 -0.1565
I ' ' ""' :;:;~;. '~;:;;;...;;::~"~:;;:~""~.:::• ' ..
,____________ ----- --"-~ •a-so -~ o.-22it2-~o; ••. 16
I
6
1
x· 141
-o.oa3a
:-0.0259
i '·
~.11R7
0.1'51o5
'·-- 7--FACTDii---7 DRIGf>iliCi.-riA[)fNGS :·--PERMUTED -SCORES 1274
-0.110{'
D. 7%6
0.1202
o. 2745
0.4824
o.r37J -n.to33 -0.1614
o.1636
8
FACTO~
8 ORIGIN~L LOADINGS
l--9--~:~TnR-
o.35'H
PEII~UTED
o. 11 o3 ·-o. onq
o1·
'
-0.1794
o.ct74
'J
·-o;1o15
. O. 36;~!..... g._Zf>?.5. .. .:: 0. 094.1_. __ _
'
i.
o.C859 __ :-:0_.0816 _:-_Q_._o~t,_5_ ____ gc~?.?!i.'t. ....-:.0..~.'!'6~---'
-< · - - - - '
--,---- ··-- - - - - - - - - - - - - - - - - - - - - - - - - ------ ---------'
SCORES 1274 X llot
o,;!434 _______().431~~~12
-o.o~o11
42:
u
0.0083
X-~~~-----
n8e:J:r~~N;L.; ~7~·4:o~~:o:.· 7o:8E~:oou·_,2:;28 :o~oes.23128754 x
oo9.;oo
,-0.1289, ____ !)_._t,QQ3•.• ::0_.1C)8,_
,:,~ ;~;;;,m ;:,;;, -:~:;;;; m:~.~... j
6 ORIGI~~L LOADINGS
PERMUTED SCORES 1274 X--141.
n.1309
0.0190 -0.1628 -0.0409
-0.3085
0.06~6
0.0148
0.0506
FACTOR
'
0.2232
s91
1-,.,-
~
-0.1618
-0.0743
0.1242
------- - -· ------------------------------------ .... ---0.2520
0.2021 .. :-:.I1_•_?0?'J.....~~-3!!_to~ •• .::~.•_2_5_4!1__ __
-o.o389
-10--- FACTOR ·1i' OR iGi ..~L LOAOINGS
J0ER-HUTED SCORES 1274 X 141
o.i992
~.,151
.o.11o36 -o·.o330 -0.1612
o.1035____o~0268 ___ Q_•J.1_4_1___.::~-~?.?9.'!. ____ ()_.;JO~t. ...
-0.2794
0.1576
0.0846
0.0744
11
I'
ACTOR~~~ O~:~ GIN~~ 3 ~~~01 ~~~13 o:ERM~~~ ~2 !COR~~ 2 :~~tt-X -f.;I--~~-~~-~~---~-~~~-;~-----~~~~-~~---=~---~:~-;-~~.-;-;~-~----O.I951o
' 12
~0.0087
0.261o5
0.1466
I;ACT(ll> 12 ORIGI"lAL LOADINGS
PEIIMUTE·Il--SCORES 1274--K 141·-------------------------------------------------0.1689
'l.::'l35
0.3118
0.3583
0.2807
0.1327
0.0571t____ 0_.1129
_0.~80_2_ ___ ().1_!30_
-0.04~9
0.2701
-0.1890 -0.0253
-~ ~---FliClrfli\'l. ·wnGI-,i&I(_(_LoiAilT~GS___ PERMli'rE !i'-SCORES-f2i~
-0.0874
1.0377
{'.2750 · -o.to33
1..
1..
-0.2323
-o;1488
. 'l.1360
-0.1106
FACTOQ 14 ORIGI~l LOAOINr.S
PEA~UTED SCORES 1274
-0.016.3
'). 3132
o. 0566
0'.1871
0.0727
0.1337 -0~1316 -0.1019
0.5916
I
x 14 t
0.2184
0.0323. -0.11>116
X
141
O.l381
__0._3236 .. ::9.·-~_1_1!! .. ::()._~~~----
64
·I
~--------------------------------------------c--------------------l1\J!Lll..~---------------------------------------------------~
r-----------------------------~----~----QlUGUlAL_.SC.ORES__MI>__£f:~l.!TI':.O_.SC.OI\i:S__:-__ l_li_O_\_RQI'I.S.JIRQ_.COI..I11mS. ______________________ ~
MODIFIED COEfFICIENT OF IN~~~----~----------------~----
'
ORIG!~Al
~---
-0,1)5')7
-O,!'il1
0,0721
-;---·F-At rnp
2
"~,
r,, NI\L
li1AriiNGS___ __ o. 0349 ____ .,-:J, oJB
PF~ ,..urEc·-·~c-cRiS--ii74--x--i4J
__________ _:('.~5('_7_ __,._), 0273
r~1747
-O,J24S
r
PERMUTfD SCD~E~ 127~ X 141
O.l'Hb
<:>.27.3'•
_(),<:1128 __________ - 0.1328- _,-_0.1980
-O,ObAZ -').150'>
LOAni~GS
l'o?. 0 1"
C~C554
I
--~
Oo143l
_-0,0646
0.1270.1
_______________________ -------- ·-- ---------
_ -,o.~9.99. ________o_.03b.L•.. 0.157 L_ o.oaz.~t______ o.oo32.
o.zz~.l___.
0,!510
~ nRJGI~~l tnAni~GS
PERMUTED ~CO~ES 127~ X 1~1
o.I'J36
IJ,CH'1 -c.1-431 --J.CB40. o.z~o3.
_-o.oozL .o.o1oo -o.o~t51
0.1376
0.1254 ..
-o.oqzh
o.25~2
-c.ce77 -o.o129
'-----:c::-=::-:---:--=- - - - - - ·- - - - - - - - - - - , - : : - , - - - , - - : : - : : - - - - - , - - , - - - - - - - - - - - ~
FACTOP 4 'JPIRI~~l .LOADINGS
PFR~UTEO SCORES 1274 X 141 ·
1- ------ -- - 0 •• ~_2 3 1 -0,1)374 -O,'JR39 -0.072'1 -0.1636 .. ------ ,-0 •. 1357. __ 0.1760 -0.1619 0,2722 -O.l21't~0 "
~
~.2z78
",09Rq
0,0851
,----5- -- FACT'lD 'I ODIGI N~l l'l~n•~c~ PEq!'uTEo ~ciiiies i.i74 __ i41 _______________ ----- ----· --- - -r----------D.Dl2R ---D. I 'l?l\. ___ o.zt.nz____,o,It.J4--D-.-U.Os___~o.ol ~5
O.cl-lll.b----O·l-llD2--0.zn 4-~o.uz.
3
1 71
~---:
!!
I,
r·
--
l'l,l'i'>R
~Aefll!!
__
1),142'1
Coi4Z2
-0.0~32
x__
_ --------------·------------------
n~lfiPJAl HIAIH'lfi§
~[Q\IIJWJ §!Hm~§ IH4
"•1!1~
1,!H1,~
•O.e>ozg -o.l35t -O,Oi3h.
tJ
1'.~54'\
-o.n34Q
o.ozq~
X 141
0.7648
-O.:H4il
.. o.a113
FACTnR 7 nRIGINAl LnAni~GS
PE"~UTEO SCORES 1274 X 141
_-,0.1979
!1.1572
0.011~0
0.1758
0.2877_____ __
0.0175
-o.aa'J~
-o.o7q7 -o.12~4
o.0435
~0.1672
-~0.0536.
ll.i81t
0.0059 -
o.1076 _ -o. Oltltl
.I
8
FACTfl~
~ noiGINAl LnAO!NGS·
PER~tJTfO SCORES 1274 X 141
- - - - - - ' - - ____ ':1.1432. ____ C!, 0923 __-(), 045 L _.,.1),16Z!L._0 •. 1SOL________Q..26!t.7_.,-0.0538 _ _,_Q,l9B 7_ -0.142.1._ .. 0. Qb6L..i
-_-_--9_.
0.;)~54
0.062_7
0.2107 -0.03~1
'
FACTOR
q
no)GI~Al l'J~OINGS
-C.D&49
-n,1B65
1-
1.0033
Oollh9
PER~UTEO
0.1376
-0,0~51
SCORES 127~ X 141
IJ.Z722 -0.2715.
-0,0743
~O--FACTORinrwr-GIN4l LO~r)j-;jGS--PfR~UHO SCCRES 1214
0.1273
-~.25Ql
11
fACTn~
0.2246
0.1543
0.1254
-O,CI24
-0.1215
X
0.181~
0.1076
-0.1~21.
0,209\
-0.2011.
141
-0.1126
0.0059
-0.0443
0.0662
-'0.2032
o.2111
0,0542
-0,0098
Oo0054
-0.18&5
.:0.2591
0.036~
11 nor~l~Al L'JAO!~GS
PER~UTEO SCORES 1274 X
-,1),!511
n,1747 -0.0~21
0,017~
0,1558
~.1141
-~.0616
0.~063O.OR54
1~1
.
·---12-,---ii\C:r-o;,---i-?--nii-iii-IN~..-t--t 'nAriirili 5- -- j;'Eii-,.;u-iicil- sccoi'E-s--,-z-i4--x--i41-------------------------------"--~--------- ------ ------- ------i .
o.on"' -'1.1247 0.2552 _ o.zz.77 __ .o.• l'tZ6 _______________ o.o.3.49... -o.o7os o.o6;;J_ o.H&a o.15U... j
.- _-_
I
---
14
~o.o63'i
FACTO~
I"\
o.zo;1~
'lR!GI~~l
-o.ror.~J
"),?163
-0.1~14
l'lAO!~GS
n.o554
-o. oa1s
-~.1814
-0.1215
·
-~.1013
1274 X 1~1
o.on«:~_____ __ ()_,llt22 _____________o.o2a?__ -0.1225
PEQMUJE~
~CORES
0.2107
-~.113~
14 noiGI~Al I.OAOI~GS
PE~"IITED SCORES 1274 X 1ltl
____..,_C. 150ft_ ~.1510 ----~0.0729 _____ 0~D.853 --=0~Da.3..L ______ Q.!)31J ____Q,Q435 ___-Q,038l._
o.o~~z
-0.1033 -0.1137
o.4569
J'ACTn~
,-,Q,07~3
65
of the cases of row and column permutation.
In this
respect, all three measures agree with the criteria for an
;adequate measure of invariance. As more rows and columns
i
!of Z were permuted, the relationships between the two sets
i
I of
i
factors was shown. to decrease by each of the measures.
i
\..
jThe expected result of permuting 100% of the rows and
I
!columns of Z to formZ* was that an adequate measure of
i
/invariance. should reflect only chance relationships between
!
I
lthe factors.
The coefficient of congruence produced the
!lowest relationships.
The coefficients of invariance were
ihigher than either of the other two measures.
Results for
I
I
!the modified coefficient ore invariance were lower than the
/coefficient of invariance but slightly higher than the
I
·!coefficients of congruence.
I
Typical Study
For each of the
i
i
case~
mentioned previously the
l
;original matrix of factor loadings was derived from data
i
!collected on the initial day of a two week Head Start
i
;
t1!
•
•
ltra1n1ng program.
The second set of factor loadings was
!computed from data collected on the last day of the
1
I
!training program.
This second set of data consisted of
[responses of the same subjects to the same attitude survey.
i
iThus, results are presented for a typical application of
i
I.
.
!the fixed-variable, fixed-subject design.
:
i
iResul·ts Based on Principal Components Loadings
I
Coefficient of Congruence.
Table 26 presents the
66
,--- ------------------ ------- --------·
6
~--
ORIGINAL (PRE) SCORES
;::~~ ~~~~5 -~--~~~~~:~~;;~:::·:·:::: :..
__ . . ......
J
~---------------------------------------CO~~F~C~~NT OF~~RU~EN~C~E~----------------------------------~~
~~~~-~-----~-~:~~~---·:··OR! r.INAL
LOADINGS
Q.8346
0.1199
o .• o7.3e ____,_o.o661o
2
fACTOR
ORI
-0.0200
o.005Z
GIBN:SLE:C::~Rll:~:~ l:MPONENTS...... ---- -------------------
-
-~
o.0427· · 0.1102
-o.oo1a
o.v7a3 -0.0211 ~o.oois
o.ozla·
-,o.oz'to. ____________________________~-------~-----
7 ORIGINAL LOADINGS
ORIGINAL SCORES IZH X. llol. .. ..
.
-o.12s6
o.s353
o.1998
o.on4
o.o365
-o.o65Z
-o~u89 -o.U56 -o.ul's··-·o.osn
0.0286 -O.Oit92
0.0038 -0.0988_
.............. ···-····---· -------· ...................... --------- .......................... ~
!
~~I;!!J.~ORIGI.NAL . LOADINGS
ORIGINAL SCOJIES 127~1!_!_4_1_______ ----------·-------------------
/ .. ___ . . . __ ~:g~~! -g: ~~~: __ g: ~~~!
g:~~~~ -o .o_8!H
____
----~:-~~-~~- _~~--036~--- -~--o-~'-~_: __ ~.o-9~-~---~~=-1-~~~__j
FACTOR
4 OI!.IGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
1
-0.0164
o·.oo95 -0.1016
o.3986 -o.244to
o.z3o7
o. llo27
~-----=-.D..022L_:,.._0.0.036 ____0 •. 015.9._0. OOZJ
--------------------,---------1
I· . 5
_·FACTOR
5 ORIGINAL LOADINGS
ORIGINAL SCORES 1214 X 141 .
. .. _
o.o,n -o.oz47· -0.3193 -0.11127
o . n n -0.09toO
.-o.o376 -o.o2n -o.0552 -o.oo4o
-o.o~95
-O.C035
~___fACIDR __ b
r-..
ORIGINAL LCADINGS
ORIGINAL SCORf$_127~_X_1~1~
-0.0567 -0.0858 -0.0174
0.0511 -0.~205
0.0714
0.0225 -0.039't -0.1986
0.1602
.
--- --
o.n82
-
7.
FACTOR
_ 8.
FACTO!!.
o.2!>35
-o.c.4117
o.ona
-O.C919
---·::1
1 ORIGINAL LCADI'IGS
ORIGINAL SCORES 1214 X HI_
0.0489
0.0652
0.0640
0.0093
0.141t3
0.0718
0.01::15
o.o1 !>3
0.0492
0.0935
i-----------"'-"0 .U.lD __ ..0.18 !l5 .... o. 0122 •.. 0. 0 380
-----------~----'-------------------------i
1
'~
8 OI!.IGINAL LCADINGS
ORIGINAL SCORES 1214 X..1'tJ:...
-0.0342 -0.0341
o.0124 -0.0591
0.2652
-0.0160 -0.0616
O.l67't
.. .............
o.o~n
0 1262
--~~-=FAC.t~l!.--~
ORIGINAL
-0.0166
-_.- -.- - -,--~-- -.--0.2010
_
•
. ........ __
0 06 6
0 14 -5·0• ·----~-~1
------------~-----------------c--~---1
O.Zb17
0.0827 -0.0166
0.0168
LOADINGS
ORIGINAL SCORES C2.Jit .X .lloL.
0.1098 -0.1888
0.0262
-0.0563
o.2351
·o.I3B6
-0.~501
-o.o5I6
-0.0628
-o.o885
FACTOR 10 ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X 141
0.0532 -0.021t7 -0.2992
-0.0545
0.0429' -0.1081
0.1231
0.1471
-0.0380 -0.0311
_.,o .oo2ll_D. ot63 __.. o. oo5.7 __ o.lus__________________________________________~
-~
FACTOR I 1 ORIGINAL LOADINGS
ORIGINAL SCORES 1274 X. UL.
o.o294 -o.o6~~ -o.o11o -0.1137 -o.0969
-0.2291
o.10o4 -o-.1432 -0.1022 ..,o.C829
- o. 0225 . a. 0651t . -0.1351 o.1933 -------------------------------------· ______________ ---------------- -------------------~
11
FACTO~ 12 ORIGINAL LOADINGS
ORIGINAL. SCORES 12H X 141 _
-0.05~2
0.00()8
0.~22
0.0003
0.1309
0.0467
0.071t5
0.0013 -0.0120 -0~0340
.. ______________ o .0439 ____ o. o 11 o. _. o. O% 1. _ ,-o. 0810_____ ..... _____ ------------------ ____________ -----------------,-- ... __________ _
12
F•CToq I ) ORIGINAL LOADINGS
ORIGlltAt SCORES t~74 lt_l~l':. ..
· ·
13
o.o706
~.0112
o.o112 -o.o225
o.o248
o.o5oo -0.0121
o.0316
i---------·-o.1tsl__ ~o •. lOol .. -~-o.25oo. __-,o •. o.l9f> _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
lit
FACTO!! 14 ORIGI'IAL LOADINGS
DIUGINAL SCORES 1274 X 141.
0.0151t -0.0578 -0.0323 ~0.0774
0.171ol
Oo1969
Oo0916
O.:J454
0.1581
0.0450 -------~- ---------
-o.oza~t
_
-0.0622
-O.C469
-0.1114
-0.1129
1
67
results of correlating the two appropriate sets of factor
scores based on principal components. loadi?gs.
These
·•
results indicate that factors I and I', II and II', and III
I
~nd
III' are highly related to each other.
The coefficients
'
I
I
;presented in Table 26. range from • 0003 to • 8346.
!
Coefficient of Invariance.
Tables 27 and 28 show the
I
.two solutions for this measure.
Inspection of both sets of
·coefficients shows that factors I and I', II and II', III
!
and III', IV and IV' and IV and V' and Vand IV' are highly
i
related to each other.
Inspection of both tables also
indicates that the coefficients range·from .0004 to .9931.
~n the typical study one solution for the coefficient of
~nvariance is not the transpose .of the other.
A compari-
1
·son of Tables ·27 and 28 reveals that although the numerii
I
i
cal values are similar, the numerical values are not the
I
pame.
i
Results Based
on .Varim.ax
Loadings
I
i
Modified coefficient of invariance.
I
Table 29 pres~nts
i
I
\the. results of this measure for the typical case when the
i
Jfactor scores are based on varimax rotation of the
i
I
principal components loadings.
f',
i
V and XIII', VI and IX', VII and III', XI and XI', and
.
~II
'
Factors I and I', IV and
and X' are highly related to each other.
.
The range of
kodified coefficients presented in Table 29 extends from
I
,,;·
1
I
0004 to • 7161.
Coefficie·nt of Congruence.
Table 30 shows the results '
. 68
...
ft
•----- --· _____ --------------------- _------ _________________ ..
•-----------------·----·-·- __ .
. .. -- j
TABLE. 27--------- -------------------------------------·'- .• :..
. ORIGINAL (PRE) SCORES AND PO. ST SCQ.BES __..,_.TYPICAL :STUDX. .. ----------. -·------. ----·-··-
~----------------------~----------------COEFFICIENT-QE.INV&BI&NCE (I)
----.-~·
.
_
BASF.D ON PRINCIPAL COMPONEH'l'S---·;••~-------------------- ••-.• l ORIGI~AL lOADI~GS
ORIGINAL SCORES 127~ X
n.9915
o.1287 -o.o776
o.o11~.
o.1196
0.0887
0.0959
0.1~3
0.1001
FACTOR
1~1
-o.o8o8
0.1776
-0.1095.
o.o~~z
o.0080
2
FACTOR . 2 OR .. GI'II&l lOAD I .. GS
CRJGI NAl. SCORES . i27~·-x-· 1.;f"· -·· -----· --·- -· --·· ---------· ·-·------. ·
_____________:-,_,0.0501
0.8881
O.<t582
0.0810
0.1101.
-O.Uill
.0.1829. -0.2583 -0.2289 -0.062~ !
0.0181
o.02<t0 -0.0137 -0.1592 .. . .
- · - - - - - . ·-·. --····· .. ··---·. ··----- --· · · - - - - - - - - 1---
I
FACTOR
J
't.
·-·-·---------------!
'\ IJRI GI~Al lOADl~GS
OR I Gl NAL SCCRES .. I 27~ .j( ii;·j·--- ..... -----·- .... -·- ...-·-- ·-·------· ....
o.on'l9 -0.36~oo
o.8515 · o.1os8 -o.o792
_ .~o-0360. :-O.U8J
o.o952
_o.22,_.z __-o.1.2!0 ... ,
-0.0310 -0.0687 -0.1871
0.1171
-----,--=F..,.A-cC-=T-cO-cR--<t--0-RIGI
... Al LOADI"4GS
CR_I_GiNAi-SC0REs-r2i~t X lltl
~
0.300<t
-o.on36
o.o5~5
-o.os66
o.6901 -o.6~<t6
o.06<t2
o.1811
o.2111
o.o926
;
5 nRIGI~AL lOADI~GS
CRIGINAL-·SCCRE·s 12H.X 1~1
..... O.O'o99 . -0.0055 :~0.56'o5 __,-O.lt293 ___ __
Oo0398 -n.~t)9 -0.1590
Oo0Z47
FACTOII
5
~·_ _ _ _ _ _ _ 0.0~6l
'·
~~---,·---·FACTOR
Ii
6 !11tiGIPlAL LCA!liPlGS
o.ooo~
0 1
HC TOR- : : : :
.-'1.0110
1
-o.1o1o
0.2U7
0.2531
0.2.031
-0.1812 __ ,
0~~:~~-~~~-. ~~9~-~- 0 ...~2~---~~-03~3~-~6~ .
ORIGINAL SCORES 1274 It 14)
o.o239
-o.oo1z
o.155~
o.o379
0 ... 917. .. ,-,0.2236
.-0.1716. -0 ... 0711
GI~:-~~~~D ~:-~-:~ ~ :Rl;,;~:~~:CCRESCZ;:-:7;-:'t;-:X:;--;l;-:~;-:·;----------------------------------1
-o.oo<o~
o.o3Zo
·-o.ri351
0.1830 . ..
-O.~ll"
0.29<17
0.11o02
O.llo31
FACTOR
II nRI!;I~Al LOAiliNGS
ORIGINAl SCORES 'iz'7" X
o.3926 ....0.12l6 ... _o_.1003... -0.0551 ____ .0.• 32~.3 ... 1
1
1.__ ---. ..
8
~~~~=~OR~~:~:::GI~::~:::~I~::~~~:RIG~:::::CO:~O:::Io
l
....... .•.•,., . '·'"' .•. ,... _,_,.,
0.1186)
-0.2553
0.3561,
0~2168.
1 ..-.-----· --- ------.............
X HI O... dM
.t-i.I
FACTOR 11
oRiGiNil-iOAo .,GS
-o.~l61
0.1513
-o.o2ss
0.281o8
·CRI Gl
o.ooo"
-0.2601
~p_.O~U -0.2891_~0~2~~1
..
_
_9.1318
....... ...... •...... ~.:.;.~ _.;,,,.;:::1
FACTOR 10 ORIGI~AL LOAOINGS
ORIGI~AL SCORES 127" X 1"1
0.01·:>'1 -0.0141 -0.1,.37
0.1511o
0.1608
-0.166"
-O.~Io2l
0.0 .. 01 -0.0583
0.2319
10
·- --·--·------ . -----
-0.0833
OoC64". -0.153" .. -O.'o665 ... ,
NAL-SCO-RES --.-z:;~-x-11.-.----------------------------------------------- ____________ .,
-o.11o6
O.lo609
-o.t~oo6
· -0.2'195
a.l528
-0.1211
-.0.2162
-o.oo6o
1'
·i·····""'~r~;m···~~~i::t'!~::r·~~il:t':~;:::··--···;_ ;.~ ·.:~;:· :;:;;;~- ·-....... ..•..... ·J
FACTOR 13 ORIGINAl LOAOI~GS
CRIGINtL SCORES 127~ X 1"1
0.0083
0.0691
0.0053
0.0887
0.0023
0.0303
0.3687 -0.2238 -0.3881 -0.1228
13
I
'·~
I
FACTO~
.
.,-0.007"
0.1217
0.01113 .. _().._0790
-----------
llo ORIGI~Al lOAOINGS
ORIGINAL SCCRES 127~ ·x l"l
_o.o::u ___ O.OOlO _ -O.OCBl. -0~0605 ___ 0.1.794___ : .. __0.360" .... Oo05J5
Oo2219
Oo1l71
0.155"
0.0903
__ 0.01160. __.,.Oo.U.H_~._illj___
69
..
.. ...... _........ ·r1\BI£ _2 s__ .. ___ .... __________________________________________________________ ~
_ ORIGINAL_ (PRE.) . SCORE!>. AND..
r---------------------------------------~C9~EE~CIENT
!'.QS~_$.CQm::s __,_.7:JJ!ICAJ..S'l'llJ)Y. ____________ -----------------------{
~
OF INYARIAHCE !z•)
BASED ON- PRlNCIPAL -COMPONENTS-----·-----------------------------------·
'·
FACTOR
z
I ORIGINIIL LOA::liNGS
PERI'UTED SGDRES 1271t X 11tl
0.9911
D.0'>87 -0.0310
0.0207
-0.0277
-0.0166
o.~161
o.o112
o.o1s1
o.0161
FA6ilrl·--7--rillii;JPiAL LriAoliiGs
____________..c-O.U!J'> ..... o. 8111~
o.Do85
o.~108
I··
3
I·
-FACToR
I
- PERMUTEo· scoRes ·,21~-
-· 0.4_002
-o.OC36
o.ons. ___:...o.o209_
o.oo21
0.0096
·x-·i.r.·.--------- ----------------· --------- ·······-········-· ·· ---·-·· _____ ,
0.0516 ___ O_.J)Jj_7_1t_______-_Qo_059_<,1_-_~.9_11_1tL..=-(1 .•J!l.~_O.O~l8
-o.o546
-0.021tl
j
1
~ 2 ~:! G ~~~~ ~~~:o '"~~ 849 ;eRii~~~ ~ 8 !to~g~o!~!" ·riio·f~~~~-~~~-~-~ 0 ~~~~~-- .. -~~-~~~~---. -~. 10 ~~--~;~~-~~~~ __ ,
0
-0.0108
-0.02'>2
-0.0779
o.051t6
---·----~---
FACTOR
4 ORIGINAL LCAOINGS
PERMUTED SCORES 1271t X 11tl
0.2128
.-0.0128
0.0891 -0.0745
0.6687 -0.5275
0.1)332
0.09'))
0.1344
0.0489
___ ___
1 6
'I
FACTOR
h---F-A_C_T_O_R
uRtGti.-AL LoADINGS
PER•·tureo- scoRes t271t x 1
,..0.192l:..__ o.l)tlt19 -0.0077_ -0.6't69 .. -:O.'t156 __
0.0246 -O.C098 -0.1078
0.0151t
6 ORIGINAL LOA'liNGS
PERMUTED SCORES 1274 X 11tl
0.0011 -0.2270
0.0450 -0.0101t
0.0455
0.164_3
-0.1424 -0.2027 -0.1919 ·-O.ll92
I
8
2
0.0683
122
-O.D535
7
-D.ozt.o
-C.2072
o.o221
0.2550
0.2352
FACTOR- : :::GI-i:~ :::.,1N:: :ERH:::: :CORES
t----~-----:
__--:0o •• 0o48!02
7
I- ·9··---FACTOR
; 13
~O.Ol1b
-o.o2o1
-O.lltB5
1274
x""f~tf
-O.l164
..
.
.. 0.5233
-0.2165
-0.1569
-0.3801
------------------------------------------<.
1 ORI-GlNAi--LCADiN(is ____ PERHUTED SCORES 12T4 X 11tl
-0.0685
0 4
0.164lt .. 0.1251t .. -O.U55... j
~-~·;_;;~~~-~-2~2-~~~~3~8~~~~~~~~~ -o_.~;j
~-----5-~--F-ACTOR ··s; _ _ _ _ _ _ c ____
0.1731
_O.It320_
0.1376
0.&999
-&.0533
-- --- ---
0.316lj
.•. -_ . ---
__ 0._5_41~ __::o.o8f>6 .. :::0._296l__ .=o•_23'i? __o_•.J_2_!_1! .
ORI~INAL
9
-0.0756
0.0701
LCA.liNGS
PERMUTED SCORES C27lt. X , ...._,.Oit33
0.1015 -0.3695 -0.181t0
~0.0731
-0.211?
0.3110
0.1764
FACTOR 10 ORIGINAL .LOAOINGS
PE~HUTED
0.5985
0.0027 . -0.0150 _:-0.05Q5___ \
SCORES C271t X lltl
1- i .-----Fic_ro_R~-~~-~f-:L :~~~;::~, ~!~~~~ :e RH~~!:~-~c-oR-:~~-;;:;;-x-·1-~1-~:_·_~-~~:- ..~:_:~-~~~- ___ :::~~-~---~-~ :-~~~~~--~:~:-~~ ]~__' .
l-.- -.. -.
11.
-0.1477
0.1366
0.0012
-0.2492
-0.1795
O.lt095
-0.1918
-0.3552
- --- --- ------FACTOR 12 ORIGIIIAL LOA01NGS
PERMUTED SCORES .1271t X lit)
-o.os9o -o.o97o
o.o9Z6
o.1o28
o.111t2
o.t380
-0.057" -0.2171
D.l089 -0.21t01t
FACTJ.R
1-. 14----
-0.0661t
0.25itl
3
n
0.1836
-0.3520
-0.221tl
-0.0068
.
-- ------------------------- -----------o.l~92
-o.o111
-D.21l5
c.~>76
0.0993
0.0987
ORl-GlNALLnioii~'Gs·---PERHUTEDSCORES 1271t X 11tt
0.0558
o.3931t
0.2019
-0.2328
0.0115
-0.4319
0.1679
-o.1211t
0.0028
o.o~o2s
:-0.0098
0.1541.
---------- ----------------.
FACTOR 14 ORIGINAL LOADINGS
PERHUTEO SCORES 1271t X litJ
o .• o3n __ .. o. O'H1 -o.o27'L__:-O.lll9 ____ o.u.ez..__________ o.~V2c ___o.oJz, ____ o .. Jo_aa . ....=o_J:!.!I.3._.-:o • .z.7.11~--o.238<j
o.t282
o.175t
o.o940
.. 1
70
·•
- -.... TABLE .2.9 ..••.• ----------------------------------------------------·--- _______ J
-- ---------- . ....
ORIGINAL (PRE I . SCORES AND __.POST__ SCO.l!ES
-~-T.YP.lCAL..S.1'UDY.•....... ---~----- ------ .. , ___ -~ ___ :. -j
____ MODIFIED_COEIT.l.CUNT_...Q,_F_I.&JNV"-lUAOJR:LJILll\t.llNH.C"'E'-----------------~BASED ON-·VARIMAX-I.QADINGS-----------------------------------------~-----------------
1
i
FACTOR
l ORIGINAL LOADINGS
PERHUTFC SCCRES 1274 X
l~l
:_ ________ ----------!'!-"- 5 ~ 3---.:-.9-· () l_ ~ ! ....-.. 9.~()-~~-'1. .. ::i'.!.!l.~!<.'! ...:-.<!&!!>.~----------Jl_._()!!_Q_i __ ;:P.~~2!1.~ ___:-Q,Q~J'!...:-.o. •.O.?.II8___ ,-() ·- ~ ~Q L
0.~780
Ool338
0.2112
-C.33C7
-----2--- --FAtfoR·-- "i- iiit 1"G1-r-i i ·ci c·li oTN ~;-s-··--PE-Ri·liiTEii--sc·ci\es···a7t,-·x·T;;r··----------------------·------------------- ------------------·
-0.04&2
0.2955
o.c2s8
-o.~z1a
C.0039
-c.3~16
O.JC49
c.c979
-0.2478
0.1020
·0.0963
-C.l696
0.0131
!
-0.2C99
1----3-- ---,,AEri'iii·--3--iiiiiiif"llii-Tciolf.i<is··--~>Tii>4ufE·iJ··scoRes-·cz·7;,-·-;n·..;r-----------------------------------------------------------,
~------ ------------=g~ H};-,-:-~ ~-~ ~~~---::g-:H~ ~-----t~i~-}----Q&~,-~ _________Q_!_?_t~J___-:Q_..!_f!()_,____Q!.3..~'-~---- .o..Q ~-~~ _____Q_._!~-"~
-i
----------------'tm: --cg:m~--:<-:1il~---'g:ml----':"-'.'------'-'""L-•-·-""-'---•·•'"---'-' ._,..,,__,_,_._., ' --j
~
FACTOR
4 ORIGINAL
LOADI~GS
PER~UTEC
·SCORES
1e1~
X
1~1
~----5·-·.--FActoii~-~5-;~·;rcl·-i~-~~f~!oTri~-!~;~-;efiM~:-;r;~·.j~~:in:i,--x··w~~-.~:-~~-----~-:~-~~-~--~~:~-~~~----~:-~~~~---~~~:~~~~--n.o4B8
!'oU07
0.5398 -c.ta'll
'
ii.\C"torr ,; OR I Iii NAL Lr AU nn.~s•"""Jiiif~\Hrt"sc"di{ts'T214')('l4i"'"'"''"W"""'"''"""'---···H•-·-"·--·e -----·- .---- .. -- ---·--·
--------------------0•.0193 -'1.0738 -0.11>1>C -0.167!j
0.0778
-0.0209
0.11172
0.1405
~
.
-o. o9 41 ~ o.-u 3..,----:iJ~ f~i;a··::a~o9iT·--------------------------------_------------------ -- -----0.7161
-- · ---- -0.1C~9
·
.
li
Q~.11!!.~?...i
l ---~----~-~~-~~-~:~~~o-~.~.~~~~~~c~.~~~~-~~:f?B_'!;_~~-~~-~~-~'!-~~-~:~_:oJf.~~--~--~-~~:-.<h.l!EL::Q_._Q.3.f?J
n 6
·c
PE lir1iifE-c··scoR-ET-12-i4-·x-·iiil-----------------------------------· ----- --------- ----0.()368
-0.2057
-C.CE45
---- ii··--~:-.;;cyiiiC -tl- iii i 11.\ i-- i ioi>i<is····
~----------~p~·~0359_
-0.0Qft5
n.0274
-0.1831
.
___o_.9_f!!I.L.:-Q,_l_?JL ..
.
-O.C642
6
-0.00~9
0.0989
~---ii·-·-i:Acro"R-~:~~~;Gi~~:-o~-;~otN:-~~-~ :ER~:::: :cciie"s 12i~ x 11.·;-··
- ·-----------·--·-
i2 n62
~ .n6
·-o~
:
I
-·---~
-0.210~
-0.3579
-0.0418
0.1690
I
2
------- ------- ------- -------- ------- - -- -- -- ---
IJ. 1 !j06
o."D45 -c.oo6C.
o.1441
-o. 2958 -o.o083
··:o. 04 83--- o. o 746··-·c,-~ 3"68·.;···-------------·---------------·----------
-O.l21_~---~0.1875 ___ :-_c.ot81
I
l!l
FhCTOR 1!' PRIGI~Al LOA~INGS· PERMUTED SCORES 1274 X 141
, .... _. ______ _
-IJ .nJ62 _ 0.0628 ___ 0.11_3 3___ :-:()._Q5_~_e____-:!l_._Q_IlLQ. ____________Q,_U!!~.--:-:Q_._o~_16.... Jh9.9'!Q ___-:O.J71o3 _____0,2Q~l__ _
-0.1994 ·
0.2738
-0.0725
Co3179
---.-.-----FA"r:r-oR"-~-~:~~!G·,·N~~~~~~o~~-~~-~:::-.,-ii~~~~g 6 fr:c·~~r~:~r~4--x-·i;;,:~-::~:---:~~~-::··-:·:~90 ~---~-:~~~~---~~~~~-;~Co.~'91
o.o553 · -C.I602
-o.o214
·
---12 · · ·-;:.;;c-roR"-Yz··n i( fG" iNA L- -L o.\o if.iii·s·----P"Eifiiu'iTc-·sc cfles-·iz·-;;,·-~c·a~l------------------ ------- '
-o.tnt.4
· o.c-qoo
0.1026
-o.o923
IJ.OIOB
O.CZ29
·-r3--~tfrflf-n·-olfrc;-r,..-llf."'ToA-o1NG-s-rvc,.uno
O.ln76
-o.o~t~6
·
-0.2370
c.3~ce
-0.1803
0.2015
.0.1591
-O.Oio84
·-o.0086
0.~678
· -0.0811
·-- ·--
Oo0076
-0.0531
-0.0232
-0.0188
1). o3z~·-··c·;rss·..-.--------------------------·--
stl:REs 1214 x
-0.0498 . C'.1975
. Col671
·------
1~1
-c.ts2c··.;;o·;~;,-e61·-·--····--·------·-----
0.1086
0.0~61
-0.0967
··----------· ·--~----
71
lfbr the typical cas~.
Ii
Factors I and I' and VII and VII*
;are highly related.for this measure.
i
Table 30 also
iindicates that the coefficients range from .0013 to .4985.
I
I
I
.!summ.ary of Results for Typical Study
!
I
The results for.factor scores based on principal
!components are as follows:
,_'I
1. · There is a larger number of highly related pre
J
and post factors for the coefficient of invariance
I
than for the coefficient of corigruence.
I
I
2.
In general, the numerical values are higher for
the coefficient of invariance than for
~
!
I
th~
coefficient of cong~uence.
i
·!
.I
I
!
invariance are not the transpose of each other as
i
was true for all five arbitrary cases.
iI
i
.!
The results for factor scores based on varimax
IIloadings
.
are as
Ii
There is a larger number of highly related pre
1.
invariance than for the coefficient of congruence.
/•
i
I
I
~allows:
and post factors for the modified coefficient of
I
i
2.
With one exception, the magnitude of the relation-:.
ship between factors which were highly related for
both the modified coefficient of invariance and
II
l
I
!
!
the coefficient of congruence was comparable.
72
·•
TABLE 30 .. ------------------ ---------------------------------- -----· ---·· -- __j
ORIGINAL (PRE) SCORES AND POST SCORES -. TYPICAL-.STUDlt---------------------------------------..!
t--~-----------·----- ·-------alEFFICIENT-OF-CONGRIIENCE
.
.
.
·
.
~
BASED ON VARIMAX l.OADINGS---------------------------------------------------· ----- --·
-~_j
--.,--.F.-.A(Tj)q--rOllTCT'IIil-ITIIIJT'iG·s- CRIGINAL SCOI\ES 1274 X 141
~.1485
0.016~ 0.1S16.__________________
0.1165
0.0846 -0.2903
0.0101
Oo025;~~
-·-· -· -- C.4985
o. t
·.;.c.
0146.-·------------------------------·-------------·
·-- ·· · ·- ·· · · ·· · ·· · --·
~56
·-c~·o7'l..-·--:;;o~·o--r'lfi
-----2 .... I' ACTOR . -2 ·o Rl Gl~ At LCADJNGY--
_ _ __,o~·~2~29q
~.3?08
o.OFil----o-;u'Yl!z
ORTGTNAr·st"CR£S-·Izr4·-x--I•n·------~----------------------------------
0.1485
-0.1 762
0.2546
-0.0483
0.0238
Oo0537
Oo0682
-0.0446
C.0508
-0.1486 I
I
:
~~~: :~~:=~
1~~~~~~~~~~~~ ::~~~~~~:~_is_~:~:_;~~~1 ~-~:~:~~~~~~~~~~~~~~~~~~~~---=~=~~~~~:~~:~::~:=~~~::~~:::~~:::~:::=::::~::~:-~~:~:::::::_~: :~::~~: _
13
• 14
I
~
FACTOR 13
CRIGINAL StCRES 1274 X 141
0.0920
C.0684
0.1401
0.042'1
Ool021
.
-0.1252
-0.0201
-O.C591
fACTOR 14 ORIGI~Al LOADI~GS
ORIGINAL SCORES 1214 X 141
0.1o;9s
').03~9
(1.0565 -0.0290
Ool·074
.
0.0202
-0.!'307
Oo0038
Ool33<;
Col812
0.1341
0.0830
0.0820
-0.0249
~RIGI'IAL
0.0~84
LCAOI~GS
-0.1569
!
i
·
J
Discussion
I
This section considers implications of the results of
!the three measures of in variance for the criteria
,
!
!specified.
The findings reviewed also. indicate the need
for modification in the criteria for an adequate measure of
invariance.
In addition to their methodological
:significance, the implications of these results should be
'of value to investigators'in selecting appropriate
1
measures of invariance.
I
.
.
· . 1Independence of Factor. Loadings
I
In the case of identical scores and procedures two
I
!sets of results were obtained.
i
'
In the first of these the
reasures were obtained for.factor scores based upon
lrincipal component factor loadings.
In each instance the
reasure met the criteria of an adequate measure of
iinvariance: i.e. , an identity matrix.
When the factor
I
sc·ores were based upon ·two sets of identical varimax
1
~actor
I
I
loadings the values in the principal diagonal for
.
;all three measures we're 1 ~ 0 indicating that the
I
I
.
corresponding factors of the two matrices are in the same
1
I
!
~rder and are perfectly related~ however, only the
I
boefficient of invariance and the coefficient of
congruence produced an identity matrix.
73
74
The modified coefficient of invariance based on identi~
I
cal varimax factor loadings showed a matrix with ones in the
principal diagonal and n6n-zero correlations in the other
Since Pinneau, Schurr and Levine (1966) maintain
;cells.
!
.
I
I
I
I
!that the modified coefficient is a direct reflection of'the
I
I
!relationship among the. loadings, question was raised concerl
'
~ing
the orthogonality of the varimax factors.
~ion
revealed that vectors.·for the set of varimax factors
Investiga-
were not independent, i.e., L' L ~ D, where Dis a diagonal
matrix.
~nee
Consequently, an adequate measure of the invari-
of factors should reflect the relationships between
'different factors as well as the perfect correlation among
~he corresponding fact6rs.
I
The results of the coefficient
i
I
'
'
·pf 1nvar1ance and the coefficient of congruence incorrectly
I
I
!suggest that the set of varimax loadings is orthogonal,
I
.
:while the modified coefficient of invariance correctly
!
indicates the· lack of independence of vectors.
The issue of a non-orthogonal matrix of varimax
loadings led to computation of the orthogonality of the·
.example of . a varimax solution presented by Harman (1960);
. I
I
\ (Cf. Appendix, Table 32) .
The results confirmed the
!findings in this study, that the.vectors aie not
Lrthogonal.
I
.
Further calculations for the other varimax
I
!solutions also reveal that the factors obtained are not
!independent.
Therefore, for each of the cases involving
ithe varimax criteria, the modified coefficient of
invariance can be expected to differ from the original
j
I
I
i
...
I
-
-
criteria, as well as from the other measures of invariance,
since it is the only measure which is sensitive
of independence of the factor
.
·•
loadi~gs.
i
\
junique Solution· ·for ·the 'Coeffi·c·ie·nt· ·of Tnvariance
As mentioned earlier, there are two equations for the
i
.
.
!coefficient of invariance; one equation is based on R11 and
I
.
'
j
.
-:the other on R22·
.
The results for the artificial cases
reveal that one set of coefficients is the transpose of the
other.
s~own
One solution can be
to be equivalent to the
f
i
,other by
•
r.evers~ng
the order in which the factor scores are
;correlated:.
"'
II
RIVz
I
'I
I
I
I
=
1
N
S*'
=
'1
N
L*-1
=
L*-1
R11
=
L*-1
L
L'
=
L*-1
L
I
s
z
I
z
L-1'
L-1'
L-1'
I
I
[10]
L.
=
I
!Equation 4, presented earlier, shows the formula for
Icomputing
.
.
RIV
without
. Z*
reversi~g
the order of the factor
scores, i.e. ,
[4]
R
IVz*
=
L*.
In the identical .scores and procedures· case L*
=
L.
If
I
-
-
76
=
Rivz•
'• In the case of permuting columns, L*
1
Substituti~g P
L
=
P
L.
for L* in equations 4 and 10 one finds
iand
=
p
L
=
p
L
;
I
i
'!Thus the coefficient o·f invari.ance may be considered
I!un1que
.
I
for the cases in whi~h Z* is equivalent to Z or a
-
ipermutation of z.
I
In the typical study one solution for the coefficient
Iof invariance is not the transpose of the other.
The fact
that the coefficient of invarian.ce does not have a unique
\solution in the typical case is shown in the following
I
/equations:
L* may ~ot be replaced by either L or P
L in
/the typical case, since the actual numerical loadings as
lwell as .their position in the new matrix change.
I
I
Riv
z
=
L*-1· L
=
L-1
J
l
land
R
IVz*
:
L*
.However, taking the inverse of Rivz yields
.Thus,
I
-
-
. 77
. Rrvz
L)-1
=
...
L*
::r:
,
i
'
!which is equal to the equation presented for Rrv
.
While
Z*
the coefficient of invariance does not have a unique
solution in the tyt>ical case, one finds that one matrix is
,_:,the inverse of the
other~
Hence an investigator who relies
I
:upon the coefficient of invariance must be prepared to
·either develop a rationale to employ one: or the other
equation, or compute.botq solutions and somehow resolve the
discrepancies between them.
Row and Column Permutation
In the case of
p~rmuting
rows and columns two unequal
sets of factor loadings are obtained.
Since L
#
L* it
is expected that an adquate measure of invariance would
not result in an identity matrix; in addition as more rows
and columns are permuted the relationship between the
factors should decrease as more rows and columns were
permuted.
Empirical results for all three measures of
invariance met the criterion; that is, none of the
measures appear to satisfy the criteria e~tablished for an
adequate measure of invariance in this
case~
A comparison of the matrices obtained for permuted
.,
rows and columns with the results of permuting either rows
or columns led to these findings:
a.
The coefficient of congruence produced the same
,/
I
-
-
78
results for this case as it did for row permutation.
b.
The coefficient of invariance and modified
coefficient of invariance showed the same results ~or this
I
:case as for ·coTumn permutation.
i
i
/
Coefficie"nt· ·of "Cong'rUen·ce.
\not modify L, L*
!equation: 1,
\. !
th~
L.
=
Since permuting rows does
By substituting P
Z for Z* in
equation for the coefficient of congruence
·!when rows of Z are permuted becomes
= N1
L-1
Z'
Z*
= N1
L-1
z
p
=
L-1 · R12
Rc
iI
I
L*-1.'
z . L-1'
L-1'.
. I'
!
I
t '1ng rows and columns results in Z*
· :Permu
I
IL*
=
.
-
P*
p
z P* and
By substitution, and recalling that P
L.
=
P'
I= p-1,
I
=
=
I
I
=
=
1
N
1
N
L-1
1 L-1
N
L-1
z
I
p
z
z
I
p
z
ZI
p
z . L-1'
R12
P*
P*
L-1'
I
.
L-1'.
I
jThus the coefficient of congruence when rows are permuted
I
lis equal to the coefficient of congruence when rows and
I
columns are permuted.
Since the results of row and column
permutation for this measure are the same values as those
·•
I
-
~
79
obtained when only the rows are permuted, the coefficient
of congruence does not reflect changes in the factor load, ings.
Consequently, the coefficient of congruence does not;
I
I.
I
11.n any way measure the invariance of the two sets of
!
)factor loadings.
Coefficient of Invariance.
The following equations
I
, :!show that the coefficient of invariance for row. and column
I
:permutatiop is equivalent to the coeffic'ient of invariance
when columns are permuted.
First consider the case of column permutation.
Equa-
:tions 7 and 8, represented here, show· the two solutions
!for the coefficient of invariance for this case; as was
I
iearlier shown one solution is the transpose of the other:
j·
i
'!
'[7]
[8]
Rrvz
L'
=
R
=
IVz*
L
=
L-1
p
-1
p
L
R'
IVz
Now consider row and column permutation.
=
and columns of Z are permuted, Z*
IL*
=
p
L.
I
P*
Z
When rows
P
and
Substituting for L* in equation 3,
. -11
•
= L' p L
!Thus, the first solution for this case is the same as
i
..
Iequa t '1.on.· 7 .
1
The equation for the other solution involves
I
I
. I
;Z*, therefore, it is necessary to more completely develop
I
I
I
ithe equations for Rrv
Z*
R
IVz*
=
1.
N
L
-1
.
Z*
. -1'
L*
•
I
-
-
80
for Z* and L*,
1
R
. =
L -1 p
N
IVZ*
Substitut~ng
1
=
N
=
N
1
Z'
L.,...l
p
z
I
L-1
p
z
I
-1
p
P*
z
P*
z
I
z
=
L
=
L -1
p
L
=
-1
L
p
L
=
L-1
p
L.
p
p
I
L-1'
L-1'
L-1'
L-1'
Rll
L-1'
L'
I
Comparing this equation with equation 8 shows that they are
identical.
Rrv
Thus, equations for
.
z
and RIV
column permutation are equivalent to RIV
z
for row and
Z*
and R
for
IVZ*
icolumn permutation ..
Modified Coefficient of Invariance.
Equation 9,
:presented earlierv shows the results of column permutation
for this measure of invariance,
[9]
L
-1
p
L
-1'
:Permuting rows and columns results in the same equation as
permuting columns since Z* is not employed in the equation
1
for the modified coefficient of invariance.
Summary and Conclusions.
of congruence yields the same
In summary, the coefficient
result~
permutation as for row permutation.
for row and column
The coefficient of
invariance and the modified coefficient of invariance
reveal the same res.ults for row and column permutation as
81
for column permutation.
It was shown earlier that an adequate measure of
invariance should reflect the fact that the factor loadings
have remained identical in row permutation but
modified by column
p~rmutation.
they are
The coefficient of con-
gruence erroneously indicates that the factors are different when the rows are permuted.
Since for this measure
row and column permutation is equivalent to row permutation, the coefficient of congruence does. not reflect the
fact that the factor loadings are becoming more and more
disparate.
iModified Coefficient of Invariance
In the case of identical scores and procedures based
ion varimax loadings the modified coefficient reflects the
!
llack of independence of the factor loadings; this was
I
~discussed in the ~ection "Independence of Factor Loadings." i
lin this case the.result did not agree with the original
lcriterion of an adequate measure of invariance.
However,
las shown earlier, the original criteria should be modi-
!
I
!fied so that it is applicable to non-orthogonal as well as
!orthogonal factors.
Thus, the criteria for an adequate
!
!measure of invariance of factors should reflect the
I
I
!relationships between different factors as well as the
!perfect correlation among corresponding factors.
Comparing
the three measures of invariance for the case of identical scores and procedures based on varimax loadings
82
'reveals that only the modified coefficient of invariance
meets the new criteria; i.e., this measure indicates that
the two sets of varimax factor loadings are identical to
each other but not orthogonal.
In the case of permuted rows the original criterion
established for an adequate measure of invariance was an
'·
identity matrix.
The modified coefficient did not conform
to this criterion; however, the same modification in the
criteria are applicable for this case as for the case of
identical scores and procedures.
The results of permuting
rows were the· same for 30%, 60% and 9·0% row permutation. In'
addition, these ·results were equal to those for identical
§COT@§ and proe@dUr@§ ba§@d on varimax loadings.
Thus, in
each case of permuted ·rows the modified coefficient
reflects the fact that the factors are not orthogonal as
well as the fact that the factor loadings are identical.
Since the values in the principal diagonal remain 1.0 for
the -row permutation case, the modified coefficient
continues to accurately reflect the fact that permuting·
rows of
z
does not· change the resulting factor loadings.
Typical Study
It will be recalled that when subjects 1 scores were
interchanged (i.e., rows permuted) the coefficient of
congruence failed to identify the fact that the factor
loadings were identical.
When variables were interchanged
. (i.e., columns permuted),. the.coefficient of congruence
failed to identify the fact
~hat
the factor loadings were
..
83
.not identical.
Thus, the coefficient of congruence does
not measure the appropriate relationship between the two
sets of factors under these conditions.
Pinneau and Newhouse (1964) contend that the coeffi-.
-I
jcient of congruence cc:mfounds the relationship between the
I
!two sets of loadings by confusing the subject and variable
I
, !differences. Indeed, the results of permuting rows and
' .I
. i
..
1columns support this contention.
Comparing the center terms in the equations for the
coefficient of congruence and the coefficient of invariance
R.
c
L* -1'.
=
L
-1
...
or
R
= L- 1 R
L *-1'
IVZ*
22
shows that the correlation matrices differ for these two
lmeasures.
The matrix R12 , employed by the coefficient of
!congruence, is comprised of both sets of standard scores,
!i.e., R
12
lmat~ix
!.
=
~
Z'
Z*.
The values which occur in this
are a function of the reliability of the variables
land of the interval between pre and post test.
I
The
i
.j
I :
i
/coefficient of invariance .employs a correlation matrix
i
I
or R > which is comprised of the intrarelationship
22
11
;among the variables obtained at the pre or post testing.
(R
j
;
I
1
Thus, the coefficient of invariance does not include a
center term which employs divergent standard score matrices.
The results in the typical study show that the
1.-
84
coefficie~t
of invariance·reveals higher relationships·
between the. factors than the coefficient of congruence.
Since the coefficient of congruence has been shown to
confound subjects and variables, the coefficient of invariance must be considered to be the more accurate measure
of the relationships between the pre and post factors in
,
the typical study.
The center term in the equation for the modified ·
coefficient of invariance is a correlation matrix which is
ian identity matrix.
I
Hence, the modified coefficients of
i
:invariance do not depend on off diagohal values.
Comparing the center t~rms for the modified coeffi-
cient of invariance and the coefficient· of invariance
shows that while the coefficient of invariance does not
confound subjects and variables it does inflate the
,,relationships between the factors by weighting the loadI
lings by off diagonal values of the correlation matrix.
I
•
!The modified coefficient is a direct reflection of the
!comparability of the two sets of factor loadings since,
I
!by employing a correlation matrix which is an identity
I
.
!matrix, it does·not modify the loadings.
The results of
I
!100%
row and column permutation support this conclusion:
I
I
.
!These results showed that while the factors were expected
f
Jto have only chance relationships to.each other, the
!coefficients of invariance were markedly higher than the
l
:modified coefficients of invariance.
The typical case illustrates another limitation of
.,
I-
the coefficient of invariance.
The five arbitrary cases
yielded unique solutions for the coefficients of invariance .
or at least solutions which were the transpose of each
other.
In the typical study the coefficients of invar-
iance are neither unique nor is one solution the transpose
of the other.
On the other hand, in the five arbitrary
:cases, as well as in the typical case, the modified
coefficient of invariance does yield a unique solution.
Thus, comparing the three measures of invariance for the
typical study reveals that the modified coefficient of
invariance is the only measure which directly reflects the
;comparability of the pre and post factor loadings.
Conclusion
Table 31 presents a summary of the analysis based on
the original criteria.
~rthogonal
Since these criteria only apply for
factors, Table 31 indicates that the coefficient
j' jof invariance is. the most adequate measure of invariance
/ )with the exception that it does not provide a unique
I !solution in the typical case.
When non-independent factors
Jdare compared the coefficient of in;ariance erroneously
I
!considers these factors to be orthogonal.
I
.
Changing the
I
!criteria such that an adequate measure of invariance
I
:'
!should reflect both the similarity of corresponding factors
j
!as well as the relationship among different factors shows
the modified coefficient of invariance to be the most
'adequate of the three.measures compared.
In addition, the
1
--
86
results of the typical study, and the 100% row and column
permutation case, indicate that the coefficient of invariance inflates the relationship between the factor loadings while the modified coefficient of invariance is a
direct measure of
\
th~s
relationship.
f'
0)
TABLE 31
A'Comparison of Several Measures of Invariance
Summary of Analysis
Predicted
Relationship
Coefficient of
Congruence
K-
Identical Scores
and Procedures (1)
Identical Scores
and Rotation (2)
I
(Identity)
T
(cosine of·
angle of
separation)
Coefficient of
Invariance
Rivl7'/.
Riv ..
Modified
Coefficient of
Invariance
R
v
I
I
T
T
":f T
< I
I
":f I
I
< I
< I
I
'
Permuted ScoresRows (3)
Column Permutation _
(4)
Row and Column
Permutation (5)
f
I
(Identity)
Relationship
Will Decrease
or Identity
Relationships
Will Decrease
Same as Columns
<I
Same as Rows
'
Typical Study (6)
< I
< I
Same as Columns Same as Columns
Not Unique
R -1
=
RIV
IVz*
z
88
·•
REFERENCES
j
I
!Harman, H. H. Modern :Fac·tor Analysis. Chicago: University
of Chicago Press, 1960.
i
\Henrys son·, S. Applicability of factor analysis in the
behavioral sciences: A Methodological Stud~ Stockholm
j
Almqvist and Wiksell,. 1957.
, I
!Kaiser, H. F. The varimax criterion for analytic rotation
i
in factor analysis. Psychometrika, 1958, ~' 187-200.
i
!Pinneau~ s. R., and Newhouse, A. Measures
i
comparability in factor analysis for
I
of invariance and
fixed variables.
Psychometrika, 1964, ~, 271-281-.
IPinne~u,
1
1
s. R., Schurr, B.~., and Levine, A. J. Coeffi·cients of factor invariance and factor similarity for
fixed variables. (in preparation) 1966.
i
.
·!Thomson, G. H. The Factorial Analysis of Human Ability.
1
(5th ed.) Boston: Houghton Mifflin Co., 1951.
i
iThurstone, L. L. Multiple Factor Analysis. Chicago:
1
University of ~hicago Press, 1947.
~rigley,
!
I
I
C. s., a~d Neuhaus, J. o. The matching of two sets •
_of factors •. Amer. Psychologist, 1955, 10; 418-419.
;
\
APPENDIX
89
.,
90
I -
I!
I
i
'
i
TABLE 32
II
"VARIMAX SOLUTION FOR EIGHT PHYSICAL VARIABLES"
I
!I
i
!
I
I
Variable
<I
Initial Solution
a.2
ajl
J
j
l~1.
I
I
I
j2.
• • • •
0
0
e
•
. 0
•
.
. ..
•
~
•
~5.
Final Solution
b.l
b.2
J
J
.830
-.396
.879
.272
.818
-.469
.919
.210
.777
-.470
.890
.182
-.401
.858
0
.786
.500
.238
.900
.672
.458
.183
' . 792
.594
.444
.135
.729
.647
.333
.250
.684
.798
..,
246
i;
·'I
:6
0
•
0
0
!
!
17.
I
•
'
8. .
I
.. .
Sum
of Crossproducts
I
.2175
:
I
I;
~dapted from Harman (1960, p.
I
I
I
I
I
I
i
Ii
I
I
304-305)
1.444